Methods and apparatuses for estimating initial target nucleic acid concentration in a sample by modeling background signal and cycle-dependent amplification efficiency of a polymerase chain reaction

ABSTRACT

Provided are methods for estimating the amount of initial target nucleic acid in a sample prior to nucleic acid amplification by polymerase chain reaction (PCR). The methods generally involve modeling signal intensity data generated across a range of PCR cycles with a phenomenological model in concert with a chemical model, to yield an estimate of the amount of initial target nucleic acid in the sample.

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

This application claims priority to U.S. Application Ser. No. 61/117,926filed Nov. 25, 2008, which is incorporated by reference herein in itsentirety including all figures and tables.

FIELD OF THE INVENTION

The invention relates to methods for estimating the amount of a targetnucleic acid (NA) in a sample prior to amplification by polymerase chainreaction.

BACKGROUND OF THE INVENTION

The following description of the background of the invention is providedsimply as an aid in understanding the invention and is not admitted todescribe or constitute prior art to the invention.

Polymerase chain reaction (PCR), is an in vitro enzymatic reaction forthe amplification of a target nucleic acid (NA). PCR is commonly carriedout as a thermally cyclical reaction; that is, a reaction tube is heatedand cooled to achieve the temperatures required at each step of areplication reaction in a single PCR cycle. After each replicationcycle, the generated NA itself becomes a template for replication insubsequent cycles. Thus, target NA in a sample grows exponentially overthe course of repeated cycles. With PCR it is possible to amplify fromas little as one or a few copies of a target NA to amounts severalorders of magnitude greater.

Quantitative PCR (QPCR), also known as real-time PCR, typically utilizesfluoresce-based detection methods to monitor the quantity of a target NAafter each amplification cycle. Various methods have been reported foranalyzing signal indicative of the amount of amplicon at each cyclegenerated from such experiments. Perhaps most common are variations ofthe threshold (Ct) standard curve method. In this method, the generatedsignal is analyzed to determine a fractional cycle number that isrelated to initial template concentration. One example of such a methodis described in U.S. Pat. No. 6,303,305 (Wittner et. al.).

Mathematical descriptions of the chemical reactions of a PCR cycle havealso been reported. For example, Stolovitzky and Cecchi, Proc. Natl.Acad, Sci. 1996, 93:12947-52 described one mathematical model of theamplification efficiency of DNA replication in PCR developed fromconsidering the kinetics of the chemical reactions involved.

SUMMARY OF THE INVENTION

The present invention provides methods for estimating the amount of atarget nucleic acid (NA) in a sample prior to amplification by PCR byanalysis of the results of the PCR process.

In one aspect, methods are provided for estimating the amount of initialtarget nucleic acid in a sample prior to nucleic acid amplification bypolymerase chain reaction (PCR). Methods of this aspect include modelingsignal intensity data generated across a range of PCR cycles with aphenomenological model in concert with a chemical model, to yield anestimate of the amount of initial target nucleic acid in the sample; andoutputting an estimate of the amount of initial target nucleic acid inthe sample to a user or computer readable format. In thephenomenological model, signal intensity is a function of background,initial target nucleic acid, and amplification efficiency defined as thefractional amount of target nucleic acid in a reaction mixture generatedin a given PCR cycle; and in the chemical model, the fractional amountof target nucleic acid in a reaction mixture generated in a given PCRcycle is estimated from kinetic or equilibrium modeling of chemicalreactions occurring in the given PCR cycle. In some embodiments of thisaspect, the phenomenological model is in accordance with formula (1):

$\begin{matrix}{z_{k} = {a + {bk} + {{Gx}_{o}{\prod\limits_{i = 1}^{k}\;( {1 + E_{i}} )}}}} & (1)\end{matrix}$

where the parameters k, z_(k), a, b, G, x₀, and E_(i) are defined asused in equations (6a) or (6b) and (1b) (in Description Section).

In some related embodiments, the background correction factors a and bare calculated by pre-analysis estimation with a model in accordancewith formula (2):z _(k) =a+bk+c(1+E _(c))^(k)  (2)

where the parameters k, z_(k), a, b, c, and E_(c) are defined as used inequation (67).

In some embodiments, the chemical model is in accordance with formula(3):P ₃ =s ² E ³−(s ²+(1−D+2q)s)E ² +q(2s+q+1)E−q ²=0  (3)

where E, s, q, and D are defined as used in equation (50) and (1b).

In some related embodiments, the background contribution to the signalintensity may be estimated concurrently with the estimation ofparameters of formula (3). In some embodiments, the estimated initialtarget nucleic acid may be directly derived from the modeling of formula(1). Alternatively, the amount of initial target nucleic acid isestimated from the initial reduced single stranded nucleic acidconcentration, s₀, according to formula (4):

$\begin{matrix}{s_{o} = {x_{o}\frac{K_{2}}{NV}}} & (4)\end{matrix}$

where s₀, x₀, K₂, N, and V are defined as used in equation (70).

Alternatively, in some embodiments, the chemical model is in accordancewith formula (5):

$\begin{matrix}{P_{4} = {{{s^{3}E^{4}} - {( {s^{3} + {( {p + {2\kappa\; q} + {2\kappa}} )s^{2}}} )E^{3}} + {( {{( {p + {2\kappa\; q} + \kappa} )s^{2}} + {( {{( {{2\kappa\; q} + \kappa} )p} + \kappa + ( {{{- \kappa}\; q} - \kappa} )^{2}} )s}} )E^{2}} - {( {{( {{2\kappa\;{qp}} + {( {{\kappa\; q} + \kappa} )\kappa\; q}} )s} + {( {{\kappa\; q} + \kappa} )\kappa\;{qp}} + {\kappa^{2}q}} )E} + {\kappa^{2}q^{2}p}} = 0}} & (5)\end{matrix}$

where E, p, s, q, and K are defined as used in equations (9c) and(15a-d).

In some related embodiments, the background contribution to the signalintensity may be estimated concurrently with the estimation ofparameters of formula (5). In some embodiments, the estimated initialtarget nucleic acid may be directly derived from the modeling of formula(1). Alternatively, the amount of initial target nucleic acid isestimated from the initial reduced single stranded nucleic acidconcentration, s₀, according to formula (6):

$\begin{matrix}{s_{o} = {x_{o}\frac{K_{1}}{NV}}} & (6)\end{matrix}$

where s₀, x₀, K₁, N, and V are defined as used in equation (69).

Alternatively, in some embodiments, the chemical model is a chemicalmodel expressible as formula (7):P ₂ =sE ²−(s+q+1)E+q=0  (7)

where E, s, q, and K are defined as used in equation (17) and (1b).

In some related embodiments, the contribution to the signal intensityfrom the background may be estimated concurrently with the estimation ofparameters of formula (7). In some embodiments, the estimated initialtarget nucleic acid may be directly derived from the modeling of formula(1). Alternatively, the amount of initial target nucleic acid isestimated from the initial reduced single stranded nucleic acidconcentration, s₀, according to formula (4):

$\begin{matrix}{s_{o} = {x_{o}\frac{K_{2}}{NV}}} & (4)\end{matrix}$

where s₀, x₀, K₂, N, and V are defined as used in equation (70).

In some embodiments of this aspect, modeling signal intensity datacomprises a nonlinear least squares curve fitting approximation methodfor estimating parameters of the model formulas. In related embodiments,the nonlinear least squares approximation method may be aLevenberg-Marquardt approximation method.

In some embodiments of this aspect, the methods further compriseidentifying a subset of signal intensity data generated across a rangeof PCR cycles for modeling. In related embodiments, identifying a subsetmay comprise identifying a range of PCR cycles beginning beforereplication is apparent and ending at a cycle where the amplificationefficiency, E_(k), has decreased to a predetermined absolute lower limitor a relative amount from the initial amplification efficiency, E₁;preferably the amplification efficiency of the ending cycle is between10% and 50% of the initial efficiency. In related embodiments, thesubset comprises the ending cycle and five to fifteen preceding cycles.

In other embodiments, the phenomenological model is expressed as formula(1); and the chemical model is expressed as a formula selected from thegroup consisting of formulas (3), (5), and (7). In some embodiments, thechemical model is expressed as formula (3); and background correctionfactors a and b are estimated in accordance with formula (67), prior toestimation of parameters of formula (3) and (7). In other embodiments,the chemical model is expressed as a formula selected from the groupconsisting of formula (5) and formula (7); and the background correctionfactors a and b are estimated concurrently with the estimation ofparameters of the formula of the chemical model.

In a second aspect, a computer program product for estimating the amountof initial target nucleic acid in a sample is provided. In embodimentsof this aspect, the computer program product is embodied on acomputer-readable medium, the computer program product comprisingcomputer code for receiving input indicative of an amount of targetnucleic acid present in a sample at multiple times during a PCRamplification; and computer code for estimating the amount of initialtarget nucleic acid in the sample; wherein, the computer code forestimating the amount of initial target nucleic acid operates inaccordance with formula (2) and a formula selected from the groupconsisting of formulas (3), (5), and (7). In some embodiments, the inputis received from a user. In other embodiments, the input is receivedfrom a device.

In a third aspect, an apparatus for estimating the amount of initialtarget nucleic acid in a sample is provided. In embodiments of thisaspect, the apparatus comprises a processor; and a memory unit coupledto the processor. In these embodiments, the memory unit includescomputer code for receiving input indicative of an amount of targetnucleic acid present in a sample at multiple times during a PCRamplification; and computer code for estimating the amount of initialtarget nucleic acid in the sample; wherein, the computer code forestimating the amount of initial target nucleic acid operates inaccordance with formula (2) and a formula selected from the groupconsisting of formulas (3), (5), and (7). In some embodiments, the inputis received from a user. In other embodiments, the input is receivedfrom a device.

The term “amount of initial target nucleic acid,” or “amount of initialtarget NA,” as used herein means the amount of target NA present in asample prior to amplification by PCR. The amount of initial target NAmay also be referred to as initial amount of target NA, or as thestarting amount of the template in the sample that is subject toamplification in the PCR reaction. Amount of initial target NA isgenerally expressed in terms of a number of copies, moles, or mass. Ifthe sample volume containing the amount of initial target NA is known,amount of initial target NA values may be expressed in various terms ofconcentration, such as copies of NA per volume, moles of NA per volume,or mass of NA per volume.

The term “polymerase chain reaction” as used herein means the well knownpolymerase chain reaction process as particularly described in U.S. Pat.Nos. 4,683,202; 4,683,195; and 4,965,188.

The term “model” as used herein refers to a mathematical or logicalrepresentation of a system of entities, phenomena, or processes.Implementation of a model may include deriving values of mathematicalparameters contained within the model. Values of model parameters may bederived by any appropriate mathematical method known in the art.

The testis “phenomenological model” as used herein in reference to a PCRprocess refers to a mathematical representation of target NAamplification with no recognition of the underlying chemical reactionsoccurring in each PCR cycle. Parameters in a phenomenological model of aPCR process relate generally to the overall PCR process, such as, forexample, amplification efficiency and background contribution to signalintensity.

The term “chemical model” as used herein refers to a mathematicalrepresentation of kinetic or equilibrium modeling of a system ofchemical reactions. Chemical models may include parameters representingvarious attributes of the system, such as concentrations of reactantsand products, and equilibrium and rate constants. In embodiments of thepresent invention, various chemical models are used to estimateamplification efficiency for PCR cycles. Chemical models of the presentinvention may include parameters such as total reduced concentrations ofpolymerase, primer, target NA, as well as equilibrium constants for thereactions that occur in a PCR cycle.

The term “amplification efficiency” or “efficiency,” E, as used hereinrefers to the fractional amount of target NA present in an amplificationreaction mixture after one or more PCR cycles that was generated in thereaction mixture by amplification. Amplification efficiency of a singlePCR cycle, also referred to as the cycle dependent efficiency, is thefractional amount of target NA present in a reaction complex after agiven PCR cycle that was generated in the reaction mixture byamplification at the given PCR cycle.

The term “about” as used herein in reference to quantitativemeasurements refers to the indicated value plus or minus 10%.

The summary of the invention described above is non-limiting and otherfeatures and advantages of the invention will be apparent from thefollowing detailed description of the invention, and from the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic illustration of an exemplary device forimplementation of a system that estimates target NA in a sample prior toNA amplification by QPCR.

FIG. 2 shows selection of a subset of PCR data for analysis from asingle profile. Specifics are discussed in detail in Example 1.

FIGS. 3 and 4 show dynamic range and linearity of QPCR data analysis byan embodiment of the present invention utilizing a two parameterchemical model (s₀), and an embodiment of the present inventionutilizing a polymerase binding amplicon competition chemical model,respectively, for samples containing genomic DNA isolated from VaricellaZoster Virus. Details are discussed in Example 2.

FIGS. 5 and 6 show dynamic range and linearity of QPCR data analysis byan embodiment of the present invention utilizing a four parameterchemical model (Gx₀), and an embodiment of the present inventionutilizing a two parameter chemical model (Gx₀), respectively, forsamples containing genomic DNA isolated from Varicella Zoster Virus.Details are discussed in Example 2.

FIG. 7 shows the correlation of HSV-1 quantitation by the thresholdstandard curve method (x-axis), and an embodiment of the presentinvention utilizing the two parameter chemical model (s₀) (y-axis).Details are discussed in Example 6.

FIG. 8 shows the correlation of HSV-2 quantitation the thresholdstandard curve method (x-axis), and an embodiment of the presentinvention utilizing the two parameter chemical model (s₀) (y-axis).Details are discussed in Example 6.

DETAILED DESCRIPTION OF THE INVENTION

Models of a PCR process useful for estimating initial target NA fromdata representative of the amount of amplicon generated in QPCR may fallinto either of two general categories: phenomenological models orchemical models. Phenomenological models include parameters relating tothe overall PCR process (i.e., target NA amplification) and generallymodel the process as a growth process with no recognition of theunderlying physical or chemical mechanisms operating in each cycle.Chemical models, on the other hand, are based on the underlyingchemistry, incorporating kinetic considerations, equilibriumconsiderations, or both, and include parameters representing attributesof the chemical mechanisms that are operating in each cycle.

Disclosed herein are methods with algorithms for estimating the initialamount of target NA in a sample from data representative of the amountof amplicon generated in QPCR. These methods may additionally be used toqualitatively assess the presence of target NA, which itself may beindicative of a disease state or condition. In preferred embodiments,the methods and algorithms of the present invention utilize a basicphenomenological model of the effect of amplification efficiency (E) andsystem background on raw, measured data (preferably fluorescence data);in concert with a chemical model of the equilibrium bindings of a primerto a single strand target NA to form a heteroduplex, and of a polymeraseto the heteroduplex to form a replication complex (i.e., chemicalmodels).

Both of the above equilibrium bindings considered in chemical modelsused in embodiments of the present invention can approach 100% at thestart of a PCR cycling process when primer and polymerase concentrationsare high and the number of target NA molecules is relatively low.However, with each successive cycle, the number of target NA moleculesincreases, primer concentration decreases, and polymerase activitypossibly declines. All three of these changes potentially reduce E(i.e., the fractional amount of target NA that is incorporated into areaction complex in a given step). Thus, chemical models employed inembodiments of the present invention estimate E for a PCR cycle based onchemical mechanistic models. Estimates of E for a PCR cycle may then beused in a phenomenological model to estimate the concentration ofinitial target NA in the sample.

While in some embodiments E is estimated for consecutive PCR cycles, itis understood that in other embodiments E may be estimated for fewerthan all PCR cycles. For example, E may be estimated for two of everythree cycles, every other cycle, or at any other sampling. Preferably, Eis estimated for at least every second cycle within the subset of cyclesanalyzed.

In preferred embodiments, the phenomenological model is a combinedPCR/background model. In some embodiments, the chemical model is a fourparameter chemical model (chemical model A). In other embodiments, thechemical model is a two parameter chemical model (chemical model B). Inyet other embodiments, the chemical model is a polymerase bindingamplicon-primer competition chemical model (chemical model C).Development of these models follows.

Basic PCR Model

Each cycle in the PCR process includes: (a) melting double-stranded NA;(b) primer and polymerase binding to form a replication complex; and (c)templated primer extension. With perfect efficiency (E=1), each cyclewould double the amount of NA, but realistic models include E as aparameter to be determined. This is because, as described above, severalcumulative factors may inhibit amplification efficiency as the number ofcompleted cycles increases. Denoting the amount of initial target NA ina sample as x₀ and target NA in the sample at the end of cycle k asx_(k), the cyclical process is fairly represented as a recursionrelating the target NA at the end of cycle k and the cycle dependentefficiency, E_(k+1), to target NA at the end of cycle k+1 (the followingcycle):x _(k+1) =x _(k) +E _(k+1) x _(k) =x _(k)(1+E _(k+1)).  (1a)

Rearranging this relationship allows for expression of the cycledependent efficiency, E_(k+1), in terms of the fractional amount oftarget nucleic acid present in a reaction mixture at the beginning andend of an amplification cycle:

$\begin{matrix}{E_{k + 1} = {( \frac{x_{k + 1}}{x_{k}} ) - 1}} & ( {1b} )\end{matrix}$

Thus, the target NA after n cycles (x_(n)) is proportional to amount ofinitial target NA (x₀) according to the following relationship:

$\begin{matrix}{x_{n} = {x_{o}{\prod\limits_{i = 1}^{n}\;( {1 + E_{i}} )}}} & ( {2a} )\end{matrix}$

This relationship can also be expressed in terms of the fractionalamount of target nucleic acid present in a reaction complex at thebeginning and end of one or more amplification cycles according to thefollowing relationship:

$\begin{matrix}{x_{n} = {x_{o}{\prod\limits_{i = 1}^{n}\;( \frac{x_{i}}{x_{i - 1}} )}}} & ( {2b} )\end{matrix}$PCR Model Dependence on Reporter Type

Introduction of a system calibration factor, R_(sys), into equation (2)provides conversion of the target NA after n cycles to net signal aftern cycles (y_(n)):

$\begin{matrix}{y_{k} = {R_{sys}x_{o}{\prod\limits_{i = 1}^{k}\;{( {1 + E_{i}} ).}}}} & ( {3a} )\end{matrix}$

In a general sense, R_(sys) accounts for the system characteristicsnecessary to relate detected signal to the amount of amplicon present atthe time of detection. In a fluorescence-based quantitation system,R_(sys) accounts for spectral attributes of reporter fluorophores andinstrument optical and detection subsystem response characteristics. Insuch systems, R_(sys) can be considered a unit conversion factor fromtarget NA amount to fluorescence detection system units.

The relationships described in equations (2) and (3a) are applicable forreal-time processes monitored by reporters that directly signal amountsof double-stranded NA such as intercalating or groove-binding dyes.However, for probes bearing an emitter-quencher pair which generatesfluorescence signal only in the annealed state or when cleaved bypolymerase 5′ exonuclease activity, the signal at cycle k isproportional to the amount of target NA at the beginning of the cycle,i.e., the amount of target NA at the end of the previous cycle. In sucha system, the relationship in equation (3a) becomes:

$\begin{matrix}{y_{k} = {R_{sys}E_{1}x_{o}{\prod\limits_{i = 2}^{k}\;( {1 + E_{i}} )}}} & ( {3b} )\end{matrix}$

or, in another form,

$\begin{matrix}{y_{k} = {R_{sys}\frac{E_{1}}{( {1 + E_{1}} )}x_{o}{\prod\limits_{i = 1}^{k}\;{( {1 + E_{i}} ).}}}} & ( {3c} )\end{matrix}$Background Model

Raw, measured signal from a PCR process may include contributions frombackground processes. Signal arising from background processes would bea systematic bias if not separated from PCR signal before analysis byany model-based algorithm. However, as estimating and subtractingbackground from raw data is itself subject to error, in some embodimentsof the present invention, a background model may be combined with a PCRmodel.

A background model that may be used in some embodiments of the presentinvention includes contributions from two terms, a and bk, representingbackground offset and constant drift after k cycles, respectively. Thus,one background model (w_(k)) that may be used in some embodiments of thepresent invention follows the equation:w _(k) =a+bk.  (4)Combined PCR and Background Models

Combining the contributions to the raw, measured signal from the netsignal (equation (3a) or (3c), depending on reporter type), and thebackground (equation (4)), leads to the following combined models:

$\begin{matrix}{{z_{k} = {a + {bk} + {R_{sys}x_{o}{\prod\limits_{i = 1}^{k}\;( {1 + E_{i}} )}}}}{or}} & ( {5a} ) \\{z_{k} = {a + {bk} + {R_{sys}\frac{E_{1}}{( {1 + E_{1}} )}x_{o}{\prod\limits_{i = 1}^{k}\;{( {1 + E_{i}} ).}}}}} & ( {5b} )\end{matrix}$

For the sake of simplicity, these two equations may be generalized byintroducing a system composite factor, G, which represents R_(sys) or

$R_{sys}\frac{E_{1}}{( {1 + E_{1}} )}$depending on reporter type used to generate the data to be analyzed. Thegeneralized model is as follows:

$\begin{matrix}{z_{k} = {a + {bk} + {{Gx}_{o}{\prod\limits_{i = 1}^{k}\;{( {1 + E_{i}} ).}}}}} & ( {6a} )\end{matrix}$

The same generalized model can be expressed without explicit use of theconcept of efficiency (variable E) as follows:

$\begin{matrix}{{z_{k} = {a + {bk} + {{Gx}_{o}{\prod\limits_{i = 1}^{k}\;\frac{x_{i}}{x_{i - 1}}}}}},} & ( {6b} )\end{matrix}$which follows from equations (1b) and (6a).

Rewriting the generalized model in this way makes it clear that use of aparticular variable or parameter for efficiency, while convenient, isnot essential to any of the presently presented models. Furthermore, as

$\frac{x_{i}}{x_{i - 1}}$is a unitless ratio of two quantity measures of identical units, ratiosof any such measures such as moles, molarities, or other concentrationscales could be used in evaluating

$\frac{x_{i}}{x_{i - 1}}.$Chemical Models

In developing chemical models for efficiency estimation, the inventorsstarted with the approach of Stolovitzky and Cecchi, Proc. Natl. Acad.Sci. 1996, 93:12947-52 and adopted some of their notation in modelingselected steps in the PCR cycle chemistry, but made simplifyingassumptions to avoid an unmanageable number of parameters. Embodimentsof the present invention include only equilibrium parameters, as oneassumption is that cycle times are sufficient that, for the subset ofcycles analyzed, reactions proceed to virtual equilibrium. A secondassumption made for some, but not all, embodiments of the presentinvention is that template concentrations are low enough that the onlyannealing process occurring is primer binding and that polymeraseactivity is constant for all cycles. A third assumption is thattriphosphosate nucleosides are in high enough concentration thatformation of a template-primer-polymerase complex results in completeprimer extension 100% of the time.

In the following description of chemical models used in embodiments ofthe present invention, single stranded nucleic acid is denoted as s,primer as p, primer-template duplex as h₀ (with the subscript indicatingthe number of nucleotides extending the primer), triphosphatenucleosides as n, polymerase by q, and reaction complex by r. The primercomprises L nucleotides, and template length is L+N. Thus, reactions ina PCR cycle (and their corresponding equilibrium expressions) includedin chemical models used in certain embodiments of the present inventioncan be expressed as follows:

$\begin{matrix}\begin{matrix} {s + p}leftharpoons h_{o}  & {\mspace{70mu}{K_{1} = \frac{\lbrack h_{o} \rbrack}{\lbrack s\rbrack\lbrack p\rbrack}}}\end{matrix} & ( {7a} ) \\\begin{matrix} {q + h_{o}}leftharpoons r_{o}  & {\mspace{65mu}{K_{2} = \frac{\lbrack r_{o} \rbrack}{\lbrack q\rbrack\lbrack h_{o} \rbrack}}}\end{matrix} & ( {7b} ) \\\begin{matrix} {r_{o} + N_{n}}leftharpoons{h_{N} + q}  & {{K_{3} \approx \infty},{\lbrack r_{o} \rbrack \approx 0.}}\end{matrix} & ( {7c} )\end{matrix}$

These lead to the expression of conservation relations amongst theproducts and reactants as follows:[s] _(T) =[s]+[h]+[r]  (8a)[p] _(T) =[p]+[h]+[r]  (8b)[q] _(T) =[q]+[r].  (8c)

In the above conservation relations, a subscripted [•]_(T) denotes atotal concentration; absence of the subscript _(T) denotes anequilibrium concentration. Integer subscripts on h and r are dropped asthe last reaction above is assumed to go to completion. That is, [r]denotes a concentration of duplex molecules with fully extended primers.

Correspondences to PCR cycles, with subscript [•]_(k) indicatingconcentrations in cycle k, are:Template at the start of cycle k: [s] _(T,k)  (9a)Template at the end of cycle k: [s] _(T,k)+[r]_(k)  (9b)Replication efficiency in cycle k: E _(k) =[r] _(k) /[s] _(T,k)  (9c)

Implementation of some embodiments of the present invention exploits thefollowing recursions:[s] _(T,k+1) =[s] _(T,k) +[r] _(k)  (10a)[p] _(T,k+1) =[p] _(T,k) −[r] _(k).  (10b)

Chemical Model A: Four Parameter Chemical Model

Equilibria (7a-c) were then solved for [h] and [r]:

$\begin{matrix}{\lbrack h\rbrack = \frac{\lbrack r\rbrack}{K_{2}\lbrack q\rbrack}} & ( {11a} ) \\{\lbrack r\rbrack = {K_{1}{{{{K_{2}\lbrack q\rbrack}\lbrack p\rbrack}\lbrack s\rbrack}.}}} & ( {11b} )\end{matrix}$

This rearrangement allows for removal of [h] from the conservationrelations in equations (8a-b):

$\begin{matrix}{\lbrack s\rbrack_{T} = {\lbrack s\rbrack + \frac{\lbrack r\rbrack}{K_{2}\lbrack q\rbrack} + \lbrack r\rbrack}} & ( {12a} ) \\{\lbrack p\rbrack_{T} = {\lbrack p\rbrack + \frac{\lbrack r\rbrack}{K_{2}\lbrack q\rbrack} + {\lbrack r\rbrack.}}} & ( {12b} )\end{matrix}$

Equation (8c) was then be used to eliminate free polymeraseconcentration ([q]) from equations (10a-b), (11b), thereby reducing themodel to a set of three equations:

$\begin{matrix}{\lbrack r\rbrack = {K_{1}{{{K_{2}( {\lbrack q\rbrack_{T} - \lbrack r\rbrack} )}\lbrack p\rbrack}\lbrack s\rbrack}}} & ( {13a} ) \\{\lbrack s\rbrack = {\lbrack s\rbrack_{T} - {\lbrack r\rbrack( {1 + \frac{1}{K_{2}( {\lbrack q\rbrack_{T} - \lbrack r\rbrack} )}} )}}} & ( {13b} ) \\{\lbrack p\rbrack = {\lbrack p\rbrack_{T} - {\lbrack r\rbrack{( {1 + \frac{1}{{K_{2}\lbrack q\rbrack}_{T} - \lbrack r\rbrack}} ).}}}} & ( {13c} )\end{matrix}$

These three equations were then further reduced to a single equationwith the substitution of E for the ratio [r]_(k)/[s]_(T,k) (as definedin equation (9c)):K ₂([q] _(T) −E[s] _(T))E[s] _(T) =K ₁ {K ₂([q] _(T) −E[s] _(T))([s]_(T) −E[s] _(T))−E[s] _(T) }×{[p] _(T) K ₂([q] _(T) −E[s] _(T))([q] _(T)−E[s] _(T))−E[s] _(T)}  (14)

Equation (14) was then further transformed by transforming molarconcentrations to reduced concentrations, s, p, and q, and including ofa ratio of association constants, κ, according to the followingequations:s=K ₁ [s] _(T)  (15a)p=K ₁ [p] _(T)  (15b)q=K ₂ [q] _(T)  (15c)κ=K ₁ /K ₂  (15d)

This allows equation (14) to be expressed as a fourth degree polynomialin E, whose coefficients are functions of four parameters (subscriptsdenoting cycle omitted for simplicity):

$\begin{matrix}\begin{matrix}{P_{4} = {{s^{3}E^{4}} - {( {s^{3} + {( {p + {2\kappa\; q} + {2\kappa}} )s^{2}}} )E^{3}} +}} \\{{( {{( {p + {2\kappa\; q} + \kappa} )s^{2}} + {( {{( {{2\kappa\; q} + \kappa} )p} + \kappa + ( {{{- \kappa}\; q} - \kappa} )^{2}} )s}} )E^{2}} -} \\{{( {{( {{2\kappa\;{qp}} + {( {{\kappa\; q} + \kappa} )\kappa\; q}} )s} + {( {{\kappa\; q} + \kappa} )\kappa\;{qp}} + {\kappa^{2}q}} )E} + {\kappa^{2}q^{2}p}} \\{= 0.}\end{matrix} & (16)\end{matrix}$

As indicated above, the concept of efficiency (parameter E) isconvenient in developing the polynomial presented above as equation(16). However, one skilled in the art would recognize that a polynomialmay be developed along the same lines but using a different independentvariable, for example molar or reduced concentrations of target NA,[s]_(T), or S, rather than E.

Chemical Model B: Two Parameter Chemical Model

A two parameter model was also developed for use in some embodiments ofthe present invention based on description of the chemical model above,with the additional simplification of treating K₁=∞. The simpler modelis still necessarily recursive as it is not expressible in closed form,but the polynomial derived for this model (i.e., the polynomialanalogous to equation (16) in the four parameter model) is a seconddegree polynomial in E with coefficients that are functions of twoparameters (subscripts denoting cycle omitted for simplicity):P ₂ =s E ₂−(s+q+1)E+q=0  (17)

Equation (17) is readily solved for E,

$\begin{matrix}{E = {\frac{s + q + 1 - \sqrt{( {s + q + 1} )^{2} - {4{sq}}}}{2s}.}} & (18)\end{matrix}$

In this model, as K₁=∞, the reduced concentration, s (expressed asequation (15a) for the Chemical Model A above) is redefined as afunction of the association constant K₂:s=K ₂ [s] _(T)  (19)

As indicated above, the concept of efficiency (parameter E) isconvenient in developing the polynomial presented above as equation(17). However, one skilled in the art would recognize that a polynomialmay be developed along the same lines but using a different independentvariable, for example molar or reduced concentrations of target NA,[s]_(T), or S, rather than E.

Chemical Model A and Chemical Model B Objective Functions and PartialDerivatives

The objective function for the recursive cycle chemistry modelsdeveloped above and their partial derivatives are defined as follows.Partial derivatives of the objective functions are included here as theyare necessary for most parameter estimation methods.

As described above, the general model described in equation (6a) can beconsidered as the sum of two models representing the background (w_(k))and net signal (y_(k)), separately.

$\begin{matrix}{z_{k} = {w_{k} + y_{k}}} & ( {20a} ) \\{w_{k} = {a + {bk}}} & ( {20b} ) \\{y_{k} = {{Gx}_{o}{\prod\limits_{i = 1}^{k}\;{( {1 + E_{i}} ).}}}} & ( {20c} )\end{matrix}$

Both equations (20b) and (20c) can be written as recursions. The modelfor the background, w_(k), becomes:w ₀ =a  (21a)w ₁ =w ₀ +b  (21b)w _(k) =w _(k−1) +b.  (21c)

The model for the net signal, y_(n), becomes:y ₀ =Gx ₀  (22a)y ₁ =y ₀(1+E ₁)  (22b)y _(k) =y _(k−1)(1+E _(k)).  (22c)

Because the background model is the same regardless of the cyclechemistry model, the derivatives of the background are easily developedas:

$\begin{matrix}{\frac{\partial w_{o}}{\partial a} = 1} & ( {23a} ) \\{\frac{\partial w_{i}}{\partial a} = {\frac{\partial w_{0}}{\partial a} = 1}} & ( {23b} ) \\{{\frac{\partial w_{k}}{\partial a} = {\frac{\partial w_{k - 1}}{\partial a} = 1}},} & ( {23c} ) \\{and} & \; \\{\frac{\partial w_{o}}{\partial b} = 0} & ( {24a} ) \\{\frac{\partial w_{1}}{\partial b} = {{\frac{\partial w_{0}}{\partial b} + 1} = 1}} & ( {24b} ) \\{\frac{\partial w_{k}}{\partial b} = {{\frac{\partial w_{k - 1}}{\partial b} + 1} = {k.}}} & ( {24c} )\end{matrix}$

As the Gx₀ term in the net signal model is a product of a systemcomposite factor and the initial target nucleic acid concentration, itis also independent of the cycle chemistry model used. Thus the partialderivatives with respect to Gx₀ are developed as follows:

$\begin{matrix}{\frac{\partial y_{o}}{\partial( {Gx}_{o} )} = 1} & ( {25a} ) \\{\frac{\partial y_{1}}{\partial( {Gx}_{o} )} = {\frac{\partial y_{0}}{\partial( {Gx}_{o} )}( {1 + E_{1}} )}} & ( {25b} ) \\{\frac{\partial y_{k}}{\partial( {Gx}_{o} )} = {{\frac{\partial y_{k - 1}}{\partial y}( {1 + E_{1}} )} = {\prod\limits_{i = 1}^{k}\;{( {1 + E_{i}} ).}}}} & ( {25c} )\end{matrix}$

Chemical Model B Objective Function and Partial Derivatives

The two parameters in equation (18), s₀ and q, represent reduced targetNA and reduced polymerase, respectively, in the PCR reaction. Reducedpolymerase concentration, q, is constant in all cycles. The recursionfor reduced target NA is developed as follows:s ₀=parameter to be estimated  (26a)s ₁ =s ₀  (26b)s ₂ =s ₁(1+E ₁)  (26c)s _(k) =s _(k−1)(1+E _(k−1)).  (26d)

The partial derivative

$\frac{\partial y_{k}}{\partial q}$is calculated recursively by the chain rule:

$\begin{matrix}{\frac{\partial y_{1}}{\partial q} = {{Gx}_{o}\frac{\partial E_{1}}{\partial q}}} & ( {27a} ) \\{\frac{\partial y_{k}}{\partial q} = {{\frac{\partial y_{k - 1}}{\partial q}( {1 + E_{k}} )} + {y_{k - 1}{\frac{\partial E_{k}}{\partial q}.}}}} & ( {27b} )\end{matrix}$

By implicitly differentiating equation (17)

$\frac{\partial E_{k}}{\partial q}$is derived:

$\begin{matrix}{\frac{\partial E_{k}}{\partial q} = {\frac{1 - E_{k}}{1 + s_{k} + q - {2s_{k}E_{k}}}.}} & (28)\end{matrix}$

The partial derivative

$\frac{\partial y_{k}}{\partial s_{o}}$also requires use of the chain rule:

$\begin{matrix}{\frac{\partial y_{1}}{\partial s_{o}} = {{Gx}_{o}{\frac{\partial E_{1}}{\partial s_{1}} \cdot \frac{\partial s_{1}}{\partial s_{o}}}}} & ( {29a} ) \\{\frac{\partial y_{2}}{\partial s_{o}} = {{\frac{\partial y_{1}}{\partial s_{o}}( {1 + E_{k}} )} + {y_{1}{\frac{\partial E_{2}}{\partial s_{2}} \cdot \frac{\partial s_{2}}{\partial s_{1}} \cdot \frac{\partial s_{1}}{\partial s_{o}}}}}} & ( {29b} ) \\{\frac{\partial y_{k}}{\partial s_{o}} = {{\frac{\partial y_{k - 1}}{\partial s_{o}}( {1 + E_{k}} )} + {y_{k - 1}{\frac{\partial E_{k}}{\partial s_{k}} \cdot \frac{\partial s_{k}}{\partial s_{k - 1}} \cdot \ldots \cdot {\frac{\partial s_{1}}{\partial s_{o}}.}}}}} & ( {29c} )\end{matrix}$

The partial derivative

$\frac{\partial s_{k}}{\partial s_{k - 1}}$is obtained from equation (26d),

$\begin{matrix}{\frac{\partial s_{k}}{\partial s_{k - 1}} = {1 + E_{k - 1} + {s_{k - 1}{\frac{\partial E_{k - 1}}{\partial s_{k - 1}}.}}}} & (30)\end{matrix}$

Substituting equation (30) into equation (29c) reveals the general,recursive expression for the partial derivative

$\begin{matrix}{{\frac{\partial y_{k}}{\partial s_{o}}\text{:}\mspace{14mu}\frac{\partial y_{k}}{\partial s_{o}}} = {{\frac{\partial y_{k - 1}}{\partial s_{o}}( {1 + E_{k}} )} + {y_{k - 1}\frac{\partial E_{k}}{\partial s_{k}}{\prod\limits_{i = 1}^{k - 1}\;{\lbrack {1 + E_{i} + {s_{i}\frac{\partial E_{i}}{\partial s_{i}}}} \rbrack.}}}}} & (31)\end{matrix}$

The partial derivative

$\frac{\partial E_{k}}{\partial s_{k}}$may be obtained by implicit differentiation of equation (17):

$\begin{matrix}{\frac{\partial E_{k}}{\partial s_{k}} = {\frac{E_{k}( {1 - E_{k}} )}{1 + s_{k} + q - {2s_{k}E_{k}}}.}} & (32)\end{matrix}$

Thus, in some embodiments of the present invention,

$\frac{\partial y_{k}}{\partial q}$may computed recursively from equations (27a-b) and (28), and

$\frac{\partial y_{k}}{\partial s_{o}}$may computed recursively from equations (29), (31), and (32).

Chemical Model A Objective Function and Partial Derivatives

Chemical model A extends chemical model B by including two additionalparameters, p₀ and κ, representing initial reduced primer concentrationand the ratio of the primer-template binding constant to the polymerasebinding to primer-template duplex constant, respectively. The reductionin primer each cycle is represented by a recursion similar to equations(26a-d):p ₀=parameter to be estimated  (33a)p ₁ =p ₀  (33b)p ₂ =p ₁ −E ₁ s ₁  (33c)p _(k) =p _(k−1) −E _(k−1) s _(k−1)  (33d)

Two additional partial derivatives are needed over those in chemicalmodel B. The partial derivative

$\frac{\partial y_{k}}{\partial\kappa}$is computed similarly to

$\begin{matrix}{{\frac{\partial y_{k}}{\partial q}\text{:}\mspace{14mu}\frac{\partial y_{1}}{\partial\kappa}} = {{Gx}_{o}\frac{\partial E_{1}}{\partial\kappa}}} & ( {34a} ) \\{\frac{\partial y_{k}}{\partial\kappa} = {{\frac{\partial y_{k - 1}}{\partial\kappa}( {1 + E_{k}} )} + {y_{k - 1}{\frac{\partial E_{k}}{\partial\kappa}.}}}} & ( {34b} )\end{matrix}$

The partial derivative

$\frac{\partial y_{k}}{\partial p_{o}}$is computed similarly to

$\begin{matrix}{{{\frac{\partial y_{k}}{\partial s_{o}}\text{:}\mspace{14mu}\frac{\partial y_{k}}{\partial p_{o}}} = {{\frac{\partial y_{k - 1}}{\partial p_{o}}( {1 + E_{k}} )} + {y_{k - 1}\frac{\partial E_{k}}{\partial p_{k}}{\prod\limits_{i = 1}^{k - 1}\;\lbrack {1 - {s_{i}\frac{\partial E_{i}}{\partial p_{i}}}} \rbrack}}}},} & ( {35a} )\end{matrix}$where the following equality is used:

$\begin{matrix}{\frac{\partial p_{k}}{\partial p_{k - 1}} = {1 - {s_{k - 1}{\frac{\partial E_{k - 1}}{\partial p_{k - 1}}.}}}} & ( {35b} )\end{matrix}$

The partial derivatives of y_(k) include partial derivatives of E_(k)with respect to all of the parameters. These partial derivatives,obtained by implicit differentiation of equation (16), are all rationalfunctions with the same denominator polynomial:

$\begin{matrix}{{denom} = {{s_{k}^{2}E_{k}^{4}} - {{s_{k}^{2}( {p_{k} + ( {2{\kappa( {1 + q} )}} ) + {s\;\kappa}} )}E_{k}^{3}} + {{s_{k}( {{\kappa( {1 + {\kappa( {q + 1} )}^{2} + {p_{k}( {{2q} + 1} )}} )} + {s_{k}( {p_{k} + {\kappa( {{2q} + 1} )}} )}} )}E_{k}^{2}} - {\kappa\;{q( {{\kappa( {{p_{k}( {q + 1} )} + 1} )} + {s_{k}( {{2p_{k}} + {\kappa( {q + 1} )}} )}} )}E_{k}} + {\kappa^{2}p_{k}{q^{2}.}}}} & (36)\end{matrix}$

The four partial derivatives are;

$\begin{matrix}{\frac{\partial E_{k}}{\partial s_{k}} = \lbrack {{3s_{k}^{2}E_{k}^{3}} - {{s_{k}( {{2( {p_{k} + {2{\kappa( {1 + q} )}}} )} + {3s_{k}}} )}E_{k}^{2}} + {{\kappa( {1 + {\kappa( {q + 1} )}^{2} + {p_{k}( {{2q} + 1} )} + {( {{\kappa( {{4q} + 2} )} + {2p_{k}}} )s_{k}E_{k}} - {\kappa\;{q( {{2p_{k}} + {\kappa( {q + 1} )}} )}}} \rbrack}{( {- E_{k}} )/{denom}}}} } & ( {37a} ) \\{\frac{\partial E_{k}}{\partial p_{k}} = {\lbrack {{s_{k}^{2}E_{k}^{3}} - {{s_{k}( {s_{k} + {\kappa( {{2q} + 1} )}} )}E_{k}^{2}} + {\kappa\;{q( {{\kappa( {q + 1} )} + {2s_{k}}} )}E_{k}} - {\kappa^{2}q^{2}}} \rbrack/{denom}}} & ( {37b} ) \\{\frac{\partial E_{k}}{\partial q} = {\lbrack {{2s_{k}^{2}E_{k}^{3}} - {2{s_{k}( {p_{k} + {\kappa( {1 + q} )} + s_{k}} )}E_{k}^{2}} + {( {\kappa + {{\kappa( {{2q} + 1} )}p_{k}} + {( {{2p_{k}} + {\kappa( {{2q} + 1} )}} )s_{k}}} )E_{k}} - {2\kappa\; p_{k}q}} \rbrack{\kappa/{denom}}}} & ( {37c} ) \\{\frac{\partial E_{k}}{\partial\kappa} = {\lbrack {{2{s_{k}^{2}( {q + 1} )} E_{k}^{3}} - {{s_{k}( {{( {{2q} + 1} )p_{k}} + {2{\kappa( {q + 1} )}^{2}} + 1 + {( {{2q} + 1} )s_{k}}} )}E_{k}^{2}} + {2{q( {{{\kappa( {q + 1} )}P_{K}} + \kappa + {( {p_{k} + {\kappa( {q + 1} )}} )s_{k}}} )}E_{k}} - {2\kappa\; p_{k}q^{2}}} \rbrack/{{denom}.}}} & ( {37d} )\end{matrix}$Polymerase Binding and Amplicon-Primer Competition Chemical Model

In yet other embodiments of the present invention, methods are basedupon chemical models developed by extending the approach outlined abovefor chemical model A and chemical model B to account for amplicon-primercompetition. Development of these extended models requires addition of afourth reaction to the list of reactions in a PCR cycle as described asequations (7a-c) above:

$\begin{matrix}{ {s + s^{*}}leftharpoons d {K_{4} = {\frac{\lbrack d\rbrack}{\lbrack s\rbrack\lbrack s^{*} \rbrack} = \frac{\lbrack d\rbrack}{\lbrack s\rbrack^{2}}}}} & (38)\end{matrix}$

In equation (37), s* represents the strand complimentary to s, and drepresents double-stranded NA. It is implicitly assumed that [s]=[s*];thus K₄ may be represented solely in terms of [d] and [s]. This requiresthe conservation relation represented in equation (8a) to be rewrittenas follows:[s] _(T) =[s]+[h]+[r]+[d].  (39)

The remaining conservation relations shown as equations (8b-c) areunchanged. As above, a subscripted [•]_(T) denotes a totalconcentration; absence of the subscript _(T) denotes an equilibriumconcentration. Integer subscripts on h and r are dropped as the reactionin equation (7c) (describing primer extension) is assumed to go tocompletion. That is, [r] denotes a concentration of duplex moleculeswith fully extended primers.

The recursions for [s]_(T,k+1) and [p]_(T,k+1) are the same as found inequations (10a-b) above.

Following the manipulations outlined above for the development ofchemical model A, a model accounting for amplicon-primer competition(represented by equations (7a-c), (38), (39), (8b-c), and (10a-b)) wastransformed to a form more easily implemented in a computer language byreducing to a single polynomial equation in cycle replication efficiency(E_(k)=[r]_(k)/[s]_(T,k)) with coefficients that are functions ofreduced reactant concentrations. The equilibrium expressions inequations (7b) and (38) were solved for [h] and [d] and conservationrelation in equation (8c) was solved for q to remove these quantitiesfrom the conservation relations for s and p:

$\begin{matrix}{\lbrack s\rbrack_{T} = {\lbrack s\rbrack + \frac{\lbrack r\rbrack}{K_{2}( {\lbrack q\rbrack_{T} - \lbrack r\rbrack} )} + \lbrack r\rbrack + {K_{4}\lbrack s\rbrack}^{2}}} & ( {40a} ) \\{\lbrack p\rbrack_{T} = {\lbrack p\rbrack + \frac{\lbrack r\rbrack}{K_{2}( {\lbrack q\rbrack_{T} - \lbrack r\rbrack} )} + {\lbrack r\rbrack.}}} & ( {40b} )\end{matrix}$

Thus, the number of parameters is reduced to five: three reducedconcentrations (s, p, and q), and two ratios of association constants(κ, and D):s=K ₁ [s] _(T)  (41a)p=K ₁ [p] _(T)  (41b)q=K ₂ [q] _(T)  (41c)κ=K ₁ /K ₂  (41d)D=K ₄ /K ₁.  (41e)

With these parameters and the definition of efficiency found in equation(9c), the conservation relations for s and p (40a-b) are rewritten as:

$\begin{matrix}{s = {{K_{1}\lbrack s\rbrack} + \frac{\kappa\;{Es}}{{\kappa\; q} - {Es}} + {Es} + {{DK}_{1}^{2}\lbrack s\rbrack}^{2}}} & ( {42a} ) \\{p = {{K_{1}\lbrack p\rbrack} + \frac{\kappa\;{Es}}{{\kappa\; q} - {Es}} + {{Es}.}}} & ( {42b} ) \\{q = {{K_{2}\lbrack q\rbrack} + {\frac{Es}{\kappa}.}}} & ( {42c} )\end{matrix}$

Multiplying the first two equilibrium expressions in equations (7a-b),substituting Es for [r], and rearranging the resulting equation yields:Es=K ₁ [s]K ₁ [p]K ₂ [q].  (43)

Equations (42b-c) were solved for K₁[p] and K₂[q], with the resultssubstituted into equation (43), and solved for K₁[s]. This result issubstituted into equation (42a) and rearranged to reduce the model to asingle equation:

$\begin{matrix}{{{\lbrack {{( {1 - E} )( {{\kappa\; q} - {Es}} )} - {\kappa\; E}} \rbrack\lbrack {{( {p - {Es}} )( {{\kappa\; q} - {Es}} )} - {\kappa\;{Es}}} \rbrack}^{2} - {\kappa\;{{E( {{\kappa\; q} - {Es}} )}\lbrack {{( {p - {Es}} )( {{\kappa\; q} - {Es}} )} - {\kappa\;{Es}}} \rbrack}} - {\kappa^{2}E^{2}{{Ds}( {{\kappa\; q} - {Es}} )}}} = 0} & (44)\end{matrix}$which expands to a sixth degree polynomial in E,

$\begin{matrix}{{{ {P_{6} = {{s^{5}E^{6}} - {( {s^{5} + {( {{3\kappa\; q} + {3\;\kappa} + {2p}} )s^{4}}} )E^{5}} + {( {{( {{3\kappa\; q} + {2p} + {2\kappa}} )s\; 4} + {( {{p\; 2} + {( {{6\kappa\; q} + {4\kappa}} )p} + {3\kappa^{2}q^{2}} + {6\kappa^{2}q} + \kappa + {3\kappa^{2}}} )s^{3}}} )E^{4}} - {( {{( {p^{2} + {( {{6\kappa\; q} + {2\kappa}} )p} + {3\kappa^{2}q^{2}} + {4\kappa^{2}q} + \kappa^{2}} )s^{3}} + {( {{( {{3\kappa\; q} + \kappa} )p^{2}} + {( {{6\kappa^{2}q^{2}} + {8\kappa^{2}q} + \kappa + {2\kappa^{2}}} )p} + {\kappa^{3} q^{3}} + {3\kappa^{3}q^{2}} + {2\kappa^{2}} + {3\kappa^{3}}} )q} + \kappa^{3} + {( {1 - D} )\kappa^{2}}} ) s^{2}}}} ) E^{3}} + {( {{( {{3\kappa\; q\; p^{2}} + {( {{6\kappa^{2}q^{2}} + {4\kappa^{2}q}} )p} + {\kappa^{3}q^{3}} + {2\kappa^{3}q^{2}} + {\kappa^{3}q}} ) s^{2}} + {( {{( {{3\kappa^{2}q^{2}} + {2\kappa^{2}q}} )p^{2}} + {( {{2\kappa^{3}q^{3}} + {4\kappa^{3}q^{2}} + {( {{2\kappa^{2}} + {2\kappa^{3}}} )q}} )p} + {\kappa^{3}q^{2}} + {( {1 - D} )\kappa^{3}q}} ) s}} ) E^{2}} - {( {{( {{3p^{2}\kappa^{2}q^{2}} + {( {{2\kappa^{3}q^{3}} + {2\kappa^{3}q^{2}}} )p}} )s} + {( {{\kappa^{3}q^{3}} + {\kappa^{3}q^{2}}} )p^{2}} + {\kappa^{3}q^{2}p}} )E} + {\kappa^{3} q^{3} p^{2}}} = 0.} & (45)\end{matrix}$

Note that, as in chemical model A and chemical model B described above,E is not itself a model parameter but is computed as a root of P₆ whosecoefficients are functions of the five parameters s, p, q, κ, and D.

As indicated above, the concept of efficiency (parameter E) isconvenient in developing the polynomial presented above as equation(45). However, one skilled in the art would recognize that a polynomialmay be developed along the same lines but using a different independentvariable, for example molar or reduced concentrations of target NA,[s]_(T), or S, rather than E.

Chemical Model C: Simplified Polymerase Binding and Amplicon-PrimerCompetition Chemical Model

The above developed polymerase binding and amplicon competition chemicalmodel can be simplified somewhat by adding the assumption that [p]_(T)is sufficiently large that [p]≈[p]_(T). This idealization reduces thenumber of model parameters by two. From the equilibrium expressions inequations (7a-b) and the conservation relations in equations (39) and(8b-c), the following two equations can be developed:

$\begin{matrix}{\lbrack s\rbrack = \frac{\lbrack r\rbrack}{{K_{1}\lbrack p\rbrack}_{T}{K_{2}( {\lbrack q\rbrack_{T} - \lbrack r\rbrack} )}}} & ( {46a} ) \\{{K_{2}\lbrack s\rbrack}_{T} = {{K_{2}\lbrack s\rbrack} + \frac{K_{2}\lbrack r\rbrack}{K_{2}( {\lbrack q\rbrack_{T} - \lbrack r\rbrack} )} + {K_{2}\lbrack r\rbrack} + {K_{2}{{K_{4}\lbrack s\rbrack}^{2}.}}}} & ( {46b} )\end{matrix}$

Applying the same definition of efficiency found in equation (9c), butdefining new reduced concentration parameters,

$\begin{matrix}{s = {K_{2}\lbrack s\rbrack}_{T}} & ( {47a} ) \\{q = {K_{2}\lbrack q\rbrack}_{T}} & ( {47b} ) \\{{D = \frac{K_{4}}{{K_{2}( {K_{1}\lbrack p\rbrack}_{T} )}^{2}}},} & ( {47c} )\end{matrix}$equations (46a-b) are reduced to a single equation,

$\begin{matrix}{{\frac{K_{4}}{{K_{2}( {K_{1}\lbrack p\rbrack}_{T} )}^{2}}{sE}^{2}} + {( {1 + \frac{1}{{K_{1}\lbrack p\rbrack}_{T}}} )( {q - {Es}} ) E} - {( {1 - E} ){( {q - {Es}} )^{2}.}}} & (48)\end{matrix}$

Noting that K₁[p]_(T)>>1 and substituting D (as defined in equation(47c)) into equation (48) effects a simplification,DsE ²+(q−Es)E−(1−E)(q−Es)²=0,  (49)which expands into a cubic polynomial in E,

$\begin{matrix}{P_{3} = {{{s^{2}E^{3}} - {( {s^{2} + {( {1 - D + {2q}} )s}} )E^{2}} + {{q( {{2s} + q + 1} )}E} - q^{2}} = 0}} & (50)\end{matrix}$

As above, note that E is not itself a model parameter, but is computedas a root of P₃ in equation (50) whose coefficients are functions of thethree redefined parameters s, q, and D (as defined in equations(47a-c)).

As indicated above, the concept of efficiency (parameter E) isconvenient in developing the polynomial presented above as equation(50). However, one skilled in the art would recognize that a polynomialmay be developed along the same lines but using a different independentvariable, for example molar or reduced concentrations of target NA,[s]_(T), or S, rather than E.

Chemical Model C Objective Function and Partial Derivatives

The objective function for chemical model C is the same as all of themodels described above expressing y_(k) as a function of the initialtarget NA amount x₀, the system factor G, and the replication efficiencyin each cycle, E_(i):

$\begin{matrix}{y_{k} = {{Gx}_{o}{\prod\limits_{i = 2}^{k}\;( {1 + E_{i}} )}}} & (51)\end{matrix}$

In this model, net signal y_(k) is computed recursively:y ₀ =Gx ₀  (52a)y ₁ =y ₀(1+E ₁)  (52b)y _(k) =y _(k−1)(1+E _(k)).  (52c)

From the polynomial in equation (50), it is apparent that E depends onthree parameters: s_(k), q, and D. The later two are constant throughall cycles, and the first is determined as a recursion beginning withs₀, the total reduced target NA in the sample:s ₀=parameter to be estimated  (53a)s ₁ =s ₀  (53b)s ₂ =s ₁(1+E ₁)  (53c)s _(k) =s _(k−1)(1+E _(k−1))  (53d)

Implementation of chemical model C differs from chemical models A and Bin that in chemical models A and B, s₀ is a model parameter that isestimated during the modeling process. However, in chemical model C, s₀is separated into two factors:

$\begin{matrix}{{s_{o} = {{x_{o}\frac{K_{2}}{NV}} = {x_{o}K}}},} & (54)\end{matrix}$where N represents Avogadro's constant, V denotes the reaction volume,K₂ is defined above, and K is defined to represent the compositequantity

$\frac{K_{2}}{NV}.$Similarly to the term Gx₀ in chemical models A and B, one of K and x₀will be an estimated parameter. Which of these two parameters isestimated depends on whether data generated from calibrator or testsamples are being analyzed.

It is possible to estimate parameter values by a nonlinear least squaresprocedure using a general numerical method for computing partialderivatives of equations (52a-c), however in most cases programming (orusing) analytic derivatives is more efficient computationally. Thepartial derivatives necessary for most parameter estimation methods arecomputed analytically:

$\begin{matrix}{\frac{\partial y_{k}}{\partial G} = {{x_{o}{\prod\limits_{i = 1}^{k}\;( {1 + E_{i}} )}} = {y_{k}/G}}} & (55)\end{matrix}$

The derivative

$\frac{\partial y_{k}}{\partial q}$is obtained recursively from equation (52a-c):

$\begin{matrix}{\frac{\partial y_{1}}{\partial q} = {{Gx}_{o}\frac{\partial E_{1}}{\partial q}}} & ( {56a} ) \\{\frac{\partial y_{k}}{\partial q} = {{\frac{\partial y_{k - 1}}{\partial q}( {1 + E_{k}} )} + {y_{k - 1}\frac{\partial E_{k}}{\partial q}}}} & ( {56b} )\end{matrix}$

The partial derivative is obtained by implicitly differentiatingequation (50):

$\begin{matrix}{{\frac{\partial E_{k}}{\partial q} = {\lbrack {{2{sE}_{k}^{2}} - {( {{2( {s + q} )} + 1} )E_{k}} + {2q}} \rbrack/{denom}}},} & ( {57a} )\end{matrix}$

withdenom=3s ² E _(k) ²−(2s ²+2(q+1−D)s)E _(k) +q(2s+q+1).  (57b)

The partial derivative with respect to D is similarly obtained:

$\begin{matrix}{\frac{\partial y_{1}}{\partial D} = {{Gx}_{o}\frac{\partial E_{1}}{\partial D}}} & ( {58a} ) \\{{\frac{\partial y_{k}}{\partial D} = {{\frac{\partial y_{k - 1}}{\partial D}( {1 + E_{k}} )} + {y_{k - 1}\frac{\partial E_{k}}{\partial D}}}},{and}} & ( {58b} ) \\{\frac{\partial E_{k}}{\partial D} = {{- {sE}_{k}^{2}}/{{denom}.}}} & ( {58c} )\end{matrix}$

The partial derivatives

$\frac{\partial y_{k}}{\partial x_{o}}\mspace{14mu}{and}\mspace{14mu}\frac{\partial y_{k}}{\partial K}$are also computed recursively:

$\begin{matrix}{\frac{\partial y_{1}}{\partial x_{o}} = {{G( {1 + E_{1}} )} + {{Gx}_{o}\frac{\partial E_{1}}{\partial x_{o}}}}} & ( {59a} ) \\{{\frac{\partial y_{k}}{\partial x_{o}} = {{\frac{\partial y_{k - 1}}{\partial x_{o}}( {1 + E_{k}} )} + {y_{k - 1}\frac{\partial E_{k}}{\partial x_{o}}}}}{and}} & ( {59b} ) \\{\frac{\partial y_{1}}{\partial K} = {{Gx}_{o}\frac{\partial E_{1}}{\partial K}}} & ( {60a} ) \\{\frac{\partial y_{k}}{\partial K} = {{\frac{\partial y_{k - 1}}{\partial K}( {1 + E_{k}} )} + {y_{k - 1}{\frac{\partial E_{k}}{\partial K}.}}}} & ( {60b} )\end{matrix}$

Additional application of the chain rule,

$\begin{matrix}{\frac{\partial E_{k}}{\partial x_{o}} = {K\frac{\partial E_{k}}{\partial s_{o}}}} & ( {61a} ) \\{{\frac{\partial E_{k}}{\partial K} = {x_{o}\frac{\partial E_{k}}{\partial s_{o}}}},} & ( {61b} )\end{matrix}$allows expression of equations (59a-b) and (60a-b) in terms of partialderivatives with respect to s₀, computed similarly to those in equations(34b) and (31).

$\begin{matrix}{\frac{\partial y_{1}}{\partial x_{o}} = {{G( {1 + E_{1}} )} + {{GKx}_{o}\frac{\partial E_{1}}{\partial s_{o}}}}} & ( {62a} ) \\{\frac{\partial y_{k}}{\partial x_{o}} = {{\frac{\partial y_{k - 1}}{\partial x_{o}}( {1 + E_{k}} )} + {y_{k - 1}K\frac{\partial E_{k}}{\partial s_{k}}{\prod\limits_{i = 1}^{k - 1}\;\lbrack {1 + E_{i} + {s_{i}\frac{\partial E_{i}}{\partial s_{i}}}} \rbrack}}}} & ( {62b} ) \\{and} & \; \\{\frac{\partial y_{1}}{\partial K} = {{Gx}_{o}\frac{\partial E_{1}}{\partial s_{o}}}} & ( {63a} ) \\{\frac{\partial y_{k}}{\partial K} = {{\frac{\partial y_{k - 1}}{\partial K}( {1 + E_{k}} )} + {y_{k - 1}x_{o}\frac{\partial E_{k}}{\partial s_{k}}{\prod\limits_{i = 1}^{k - 1}\;\lbrack {1 + E_{i} + {s_{i}\frac{\partial E_{i}}{\partial s_{i}}}} \rbrack}}}} & ( {63b} )\end{matrix}$

Finally, implicit differentiation of equation (50) can be used to obtainthe partial derivative of efficiency with respect to reduced targetamount,

$\begin{matrix}{\frac{\partial E_{k}}{\partial s_{k}} = {{E_{k}\lbrack {{2{sE}_{k}^{2}} - {( {{2( {s + q} )} + 1 - D} )E_{k}} + {2q}} \rbrack}/{{denom}.}}} & (64)\end{matrix}$Implementation of Chemical Model A or Chemical Model B

Implementation of an embodiment of the present invention utilizingchemical model A or chemical model B begins by identifying the PCR cyclewhere efficiency was greatest, and from that point defining the subsetof cycles to be analyzed. Equation (6a) is applicable to raw data for arange of cycles beginning before replication is apparent and ending at acycle where E_(k) has decreased to a predetermined absolute lower limitor a relative amount from the peak efficiency. For example, the endingcycle may have an efficiency within the range of 10% to 50%, inclusive,of the peak efficiency. In embodiments of the present inventionutilizing either chemical model A or chemical model B, the efficiency ofthe selected ending cycle is preferably within the range of 10% to 35%,inclusive, of peak efficiency; preferably within the range of 15 to 30%,inclusive, of peak efficiency; preferably within the range of 20% to25%, inclusive, of peak efficiency; preferably about 20% of peakefficiency. In alternative embodiments, an absolute efficiency cut-offis used. In these embodiments, the selected ending cycle has anefficiency that is some predetermined percentage of the theoreticalmaximum of 100%; preferably, the selected ending cycle has an absoluteefficiency within the range of 10% to 50%, inclusive; preferably withinthe range of 10% to 35%, inclusive; preferably within the range of 15 to30%, inclusive; preferably within the range of 20 to 25%, inclusive;preferably about 20%. The ending cycle may be determined empirically bygoodness of fit of the model, or precision of parameter estimates. Inall embodiments, the subset of cycles to be analyzed comprises theending cycle and a number of cycles immediately preceding; preferablytotaling 5 to 15 cycles; more preferably, the subset of cycles to beanalyzed comprises the ending cycle and the preceding nine cycles. Theraw data for cycles within the subset may then be normalized to give asignal range from zero to one.

Equation (6b) is applicable to raw data for a range of cycles beginningbefore replication is apparent and ending at a cycle where the ratiox_(i)/x_(i−1) approaches a predefined value.

Preferably, parameters of the combined PCR/background model fromequations (6a) and (6b) and parameters of a chemical model, such aschemical model A or B, may then be estimated to fit the data in thesubset of cycles analyzed by any suitable method known in the art.Parameters may be estimated by the Levenberg-Marquardt nonlinearminimization algorithm which utilizes nonlinear least squares (NLLS)curve fitting. However, other suitable methods of deriving parameterestimates may include methods such as maximum likelihood (ML) orBayesian methods.

Regardless of estimation method employed, nonlinear parameter estimationmethods generally require starting estimates for the parameters.Estimates for the parameters G, x₀, and s₀ are made by temporarilytreating an early subset of the data as adequately represented by aconstant efficiency model. In this case, a first estimate of efficiencyE can be made from the net data, y_(k):

$\begin{matrix}{E_{o} = {{\max\lbrack {\frac{y_{k - 1} + y_{k} + y_{k + 1}}{y_{k - 2} + y_{k - 1} + y_{k}} - 1} \rbrack}.}} & ( {65a} )\end{matrix}$

Letting k_(max) denote the cycle index corresponding to E₀, the quantityGx₀ is estimated as:Gx ₀ =y _(k max)(1+E ₀)^(−k) ^(max) .  (65b)

For calibration samples, x₀ is known, and G is estimated from thequantity Gx₀ estimated in equation (65b) and the known quantity of x₀by: G=Gx₀/x₀. For test samples (with unknown x₀) G is known, and x₀ isestimated from the quantity Gx₀ (estimated in equation (65b)) and theknown quantity of G by: x₀=Gx₀/G. With an estimated value for x₀, anestimate for s₀ is obtained from formulas (69) or (70) by assumingvalues of 10⁹ or 10⁷ for K₁ or K₂, respectively, and applying the knownreaction volume, V.

Two assumed initial reduced concentrations (primer p₀=100, andpolymerase q₀=20) are justified by standard practice in PCR.Additionally, the association constant for primer binding to template isassumed to be much greater than for polymerase binding totemplate-primer complex (κ=100). With these starting values, thepolynomial of the chemical model employed (preferably chemical models A,B, C, or the polymerase binding amplicon-primer competition model) maybe solved for E by any method known in the art. For example, apolynomial may be solved for E by Newton root-finding (restricting thedomain to [0, 1]) and solving for the smallest root. Thus, E is notitself a parameter derived by the above described NLLS estimation.

Initial estimates for the background parameters a and b from equation(6a) or (6b) are readily made from early cycles before replication isapparent, for example typically cycles 3-8, and can be derived by anymethod known in the art. These initial estimates may then be used in thecombined modeling of the general PCR/background model in equation (6a)or (6b), and a chemical model, such as chemical model A or B.

Alternatively, the background parameters in equation (6a) or (6b) may beestimated separately from the chemical model parameters in apre-analysis step. Implemented in this way, a constant efficiencygeometric growth model may be assumed in the pre-analysis to estimate byNLLS the background parameters a and b along with the parameters c andE_(c):z _(k) =a+bk+c(1+E _(c))^(k)  (67)

An estimate of a constant amplification efficiency, E_(c), may bedeveloped from the ratio of a 3-point moving average offset by onecycle. That is, representing the conditioned data as y_(k), the estimateof E_(c) may be computed as:

$\begin{matrix}{E_{c} = {\max\lbrack {\frac{y_{k - 1} + y_{k} + y_{k + 1}}{y_{k - 2} + y_{k - 1} + y_{k}} - 1} \rbrack}} & (68)\end{matrix}$

It is important to note that the constant amplification efficiency,E_(c), is only used in the pre-analysis determination of backgroundparameters in equation (67). In the chemical models of the presentinvention, amplification efficiency is not constant.

Implementation of Chemical Model C

In some embodiments utilizing chemical model C, the selected endingcycle has an efficiency that is some predetermined percentage of thepeak efficiency; preferably between 10 and 50%; preferably between 20%and 40% of the peak efficiency; preferably about 30% of the peakefficiency. In preferred embodiments, an absolute efficiency cut-off isused. In these embodiments, the selected ending cycle has an efficiencythat is some predetermined percentage of the theoretical maximum of100%; preferably, the selected ending cycle has an absolute efficiencybetween 10% and 50%; preferably between 20% and 40%; preferably about30%. Selection of a subset for analysis from a single profile isdemonstrated in FIG. 2 and discussed in detail in Example 1, below.

The preferred implementation of chemical model C differs from theimplementation of chemical models A and B in that the alternative methoddescribed above (i.e., separate estimation of background parameters in apre-analysis step) is preferred. This is because estimating parametersrepresenting amplicon competition is more effective when multiplesamples with known target NA amounts (i.e., calibrators) are analyzed.In modeling calibrators, the objective function is preferably applied toselected subsequences of cycles from multiple profiles at once, ratherthan a subset of cycles from a single profile. This approach has theadvantage that parameters representing system gain and bindingconstants, which are theoretically the same for all samples, are onlyestimated once for all profiles rather for each profile independently.However, this approach still necessitates estimation of backgroundparameters individually for each sample, as background parameters may bedifferent for each sample.

Estimation of Initial Target NA from Estimated Parameters

One feature of some embodiments of the present invention is that twoindependent means of estimating the initial amount of target NA in asample are provided by the models described above. First, initial targetNA, x₀, may be directly solved from the modeling of equation (6a) or(6b).

Second, as defined in equations (15a) and (19), the s₀ parameter isdirectly proportional to the amount of initial target NA, with theproportionality factor being an association constant, K₁ or K₂ dependingon the chemical model, divided by Avogadro's number (N) times the samplevolume (V):

$\begin{matrix}{s_{o} = {x_{o}\frac{K_{1}}{NV}}} & (69) \\{s_{o} = {x_{o}\frac{K_{2}}{NV}}} & (70)\end{matrix}$

Some embodiments of the present invention provide a target NA estimationapparatus comprising a generic or specialized computer or anothersuitable logic device, such as a microprocessor or ASIC configured toexecute processes, such as software methods or other computer-readableinstructions. Referring now to FIG. 1, an exemplary device forimplementation of a target NA estimation method in accordance withembodiments of the present invention is schematically illustrated. Thetarget NA estimation device 100 includes a processor 110 configured toexecute a process, such as a software method or other computer-readableinstructions. The processor 110 is coupled to an input module 120configured to receive input from a user or device. The user may inputdata using, for example, a keyboard or other input interface. A devicefor inputting data may be, for example, an analytical instrument, suchas a PCR system for generating fluorescence data indicative of amount oftarget NA present in a sample.

The processor 110 is also coupled to an output module 130 configured tooutput information to a user. In various embodiments, the output module130 is configured to output information through a screen, monitor,printer, or means for writing computer readable information to arecordable media, or other means.

The target NA estimation apparatus 100 also includes a memory unit 140coupled to the processor 110. The memory unit 140 may be configured tostore data, software applications, or other such information. In oneembodiment, the memory unit 140 has a program product stored thereon,and the processor 110 may access the program product to execute theprogram product.

In related embodiments, the computer or logic device may be attached toa display apparatus, such as a display monitor, to display the resultsof the calculations, and/or attached a means for writing results of thecalculations to a recordable media. In these embodiments, results of theQPCR data analysis may be accessed by the operator.

The present invention finds applications in a number of differentfields, including assays, investigating differences in gene expression,gene quantitation, genotyping, investigation of mutations, gene therapy,investigation of viral and bacterial loadings, and indeed any type ofquantitative PCR analysis. For example, the disclosed methods may beused to qualitatively assess the presence of target NA, which itself maybe indicative of a disease state or condition, particularly toqualitatively assess the presence of an infectious agent (either viralor bacterial) in any clinical sample type suitable for PCRamplification.

As seen in the following examples, methods which utilize thephenomenological model in equation (6a) or (6b) in concert with eitherchemical model B (using s₀), or chemical model C are preferred. Theinventors suspect that the relative increase in inaccuracy demonstratedby the phenomenological model in equation (6a) or (6b) in concert withchemical model A may result from the NLLS method used to estimate modelparameters, which may be finding local, rather than global, minima, thusleading to inaccuracies in the estimated amounts of initial target NA.Chemical models B and C, which have fewer parameters to estimate, may beless susceptible to such influences. The following examples demonstratethat estimations of the amount of initial target NA by the methodsherein are accurate across a wide range, down to at least about 1000copies/μl. For some assays, this range may extend to even lower levelsof copies/μl, and be more accurate than obtainable by other methods.

In the specific examples discussed below, methods of the present inventwere utilized to analyze fluorescence signal obtained from QPCRanalysis. However, the above description of the present invention andthe examples discussed below are not intended to limit the generalapplicability of the invention as a whole. For example, signals otherthan fluorescence signal obtained from fluorescent dye can be used asbasis for the analysis, as long as the signal is representative of theamount of amplicon.

EXAMPLES Example 1 Subset of Cycles to be Analyzed

As can be seen in the exemplary profile demonstrated in FIG. 2, the peakamplification efficiency occurs at cycle 31. The amplificationefficiency for this cycle is approximately 0.85, and from this pointforward, the amplification efficiency decreases. The final cycle of thesubset to be analyzed is the last cycle with amplification efficiencyexceeding a preselected amplification efficiency limit. In this example,the final cycle is selected to be the last cycle with absoluteamplification efficiency exceeding 30%. As seen in FIG. 2, cycles aftercycle 37 demonstrate less than 30% absolute efficiency. Thus, cycle 38is selected as the final cycle in the subset to be analyzed.

The subset of cycles to be analyzed is then defined as the final cycleand some number of preceding cycles. As seen in FIG. 2, nine precedingcycles have been selected to complete the subset of cycles to beanalyzed in this example, giving a subset of cycles 28-37.

Example 2 Quantitation Dynamic Range and Linearity Studies

Genomic DNA was isolated from Varicella Zoster Virus (VZV, ATCC VR-1367)using a QiaAmp viral RNA manual kit (Qiagen). The isolated DNA wasdiluted in a buffer composed of Tris and EDTA (commonly known as TEbuffer) to generate a 10-E2, 10-E3, 10-E4, 10-E5, 10-E6, 10-E7, and10-E8 folds dilution series. The diluted DNA samples were amplified bySimplexa VZV real-time PCR assay on ABI7500 system (AppliedBiotechnologies, Inc.).

The Simplexa VZV real-time PCR assay used a primer mix containing VZVand internal control (IC) primer pairs. The concentration is 3000 nM forVZV primer pair and 1000 nM for IC primer in the primer mix. Thesequences of the primers were listed in the Table 1.

TABLE 1 VZV and internal control (IC) sequences VZV Dx2Quencher-AGCGGAGTGAAACGGTACAAACTCCGCT Scorpion™ (SEQ ID NO: 1)-FAM-GTTATTGTTTACGCTTCCCGCTGAA (SEQ ID NO: 2) VZV ReverseGCCCGTTTGCTTACTCTGGATAA (SEQ ID NO: 3) IC Dx2 Quencher-TGCGAACTGGCAAGCTScorpion™ (SEQ ID NO: 4) -CFR610-ATTCGCCCTTTGTTTCGACCTA (SEQ ID NO: 5)IC Reverse CCGACGACTGACGAGCAA (SEQ ID NO: 6)

The assay used a master mix containing enzymes, buffers, and dNTPsassembled according to Table 2.

TABLE 2 Master Mix Composition Master Mix Final Reaction ComponentConcentration Concentration Tris-HCl, pH 8.3 100 mM 50 nM MgCl₂ 5 mM 2.5mM KCl 20 nM 10 nM (NH₄)₂SO₄ 10 mM 10 mM dNTPs (U, A, G, C) 400 μM 200μM FastStart DNA Polymerase (Roche) 4 U 2 U

PCR reactions consisted of 12.5 μL master mix, 2.5 μL primer mix, 10.0μL sample or control to make 25.0 μL (see Table 3). Master mix, andprimer mix were added in a template-free room; sample or control wasadded in a template-allowed room.

TABLE 3 Amplification Reaction Mix Component Volume (μL) Master Mix 12.5Primer Mix 2.5 Sample or Control 10 Total 25

Amplification was carried out in an ABI7500 real-time PCR instrument(Applied Biosystems Inc., Foster City, Calif.) with the thermal programof 95° C. for 10 min for one cycle, 95° C. for 15 sec and 60° C. for 35sec for 45 cycles. VZV signal (FAM) was collected from channel A, and ICsignal (CFR610) was collected from channel Don the ABI7500 real-time PCRinstrument. Signal intensity data was collected for eight replicates ofeach diluted DNA sample.

The dilution series was used to demonstrate the linearity of amounts ofinitial target NA estimated by methods of the instant inventionutilizing the phenomenological model in equation (6a) in concert withchemical model A (P/A Method), chemical model B (P/B Method), andchemical model C (P/C Method). For comparison, signal intensity data wasalso analyzed by the threshold (Ct) standard curve method (the ThresholdMethod).

Data from the resulting analysis by the Threshold Method, the P/B Methodwith initial target NA derived from the s₀ parameter, and the P/C Methodare shown below in Table 4.

TABLE 4 Results of estimation by Threshold Method, the P/B Method (usingthe s_(o) parameter), and the P/C Method Threshold Method P/B Method(from s_(o)) P/C Method Actual Mean Mean Mean Initial Predicted Error CVPredicted Error CV Predicted Error CV Target NA Value (%) (%) Value (%)(%) Value (%) (%) 100000000 120288232 20 5 126983479 27 19 117120050 178 10000000 10827560 8 5 9120602 −9 9 9042280 −10 7 1000000 987247 −1 5951029 −5 28 881304 −12 7 100000 91813 −8 12 88071 −12 17 94626 −5 1310000 8307 −17 7 8827 −12 9 9839 −2 4 1000 830 −17 32 1111 11 10 1104 1014 100 78 −22 36 122 22 47 124 24 47

Plots of the linearity of amount of initial target NA estimated by theP/B Method (from s₀) and the P/C Method are shown in FIGS. 3 and 4,respectively. As FIG. 4 demonstrates, the P/B Method for estimatinginitial target NA from s₀ demonstrates linearity across six orders ofmagnitude.

Amount of initial target NA was also estimated with the P/A Method (fromGx₀) and the P/B Method (from Gx₀). Data resulting from theseestimations are shown in Table 5. Plots of the linearity of amount ofinitial target NA estimated by the P/A Method (from Gx₀) and the P/BMethod (from Gx₀) are shown in FIGS. 5 and 6, respectively.

TABLE 5 Results of estimation by P/B Method and P/A Method, both usingthe Gx_(o) parameter P/A Method (from Gx_(o)) P/B Method (from Gx_(o))Actual Mean Mean Initial Predicted Error CV Predicted Error CV Target NAValue (%) (%) Value (%) (%) 100000000 104575619 5 30 123582684 23 3710000000 9036335 −10 8 9642491 −4 7 1000000 1032696 3 53 1109437 11 58100000 125534 26 11 85512 −15 20 10000 9055 −10 32 8090 −19 17 1000 167267 54 1130 13 14 100 110 10 84 127 27 47

The linearity demonstrated by the Threshold Method was very comparableto that demonstrated by the P/A, P/B, and P/C methods.

Example 3 Quantitation for Cytomegalovirus (CMV) DNA

Quantified CMV (strain AD169) genomic DNA (Advanced Biotechnologies,Inc.) was diluted in TE buffer to 5, 50, 500, 5,000, and 50,000copies/reaction. The diluted DNA samples were amplified by Simplexa CMVreal-time PCR assay on ABI7500 system (Applied Biosystems, Inc.).

The Simplexa CMV real-time PCR assay used a primer mix containing CMVand internal control (IC) primer pairs. The concentration is 5000 nM forthe CMV primer pair and 1000 nM for the IC primer pair in the primermix. The sequences of the primer were listed in the Table 6.

TABLE 6 CMV and IC sequences CMVUL11M5Quencher-aggcgtgCAGCACCAACACGTGGCTACA- Scorpion™ CACGCCT (SEQ ID NO: 7)-FAM-TAACGATTACGACCGCTAAAACC (SEQ ID NO: 8) CMVUL11R1CAGCGGAAACACCGTTACAA Reverse (SEQ ID NO: 9) IC Dx2Quencher-TGCGAACTGGCAAGCT Scorpion™ (SEQ ID NO: 4)-Q670-ATTCGCCCTTTGTTTCGACCTA (SEQ ID NO: 5) IC ReverseCCGACGACTGACGAGCAA (SEQ ID NO: 6)

PCR reactions consisted of the same as described in Example 2.

Amplification was carried out by the same method and instrument asdescribed in Example 2. CMV signal (FAM) was collected from channel A,and IC signal (Q670) was collected from channel E on the ABI7500real-time PCR instrument. Signal intensity data was collected for sixreplicates of each diluted DNA sample.

The dilution series was used to demonstrate the linearity of amounts ofinitial target NA estimated by the P/B Method (from s₀). For comparison,signal intensity data was also analyzed by the Threshold Method.

A comparison of the accuracy and reproducibility of Threshold Method andthe P/B Method (from s₀) for samples with 50,000 copy numbers/μl isfound in Table 7, and a comparison for samples with 500 copy numbers/μlis found in Table 8.

TABLE 7 Quantitation of CMV at 50,000 copies/μl Threshold P/B Method n =6 Method (from s_(o)) Mean 60,386 51,144 SD 7,503 4,420 CV 12 9 (%)

TABLE 8 Quantitation of CMV at 500 copies/μl Threshold P/B Method n = 6Method (from s_(o)) Mean 467 395 SD 67 65 CV 14 17 (%)

The linearities of amount of initial target NA estimated by theThreshold Method and the P/B Method (from s₀) for CMV were verycomparable to that demonstrated in Example 2.

Example 4 Quantitation for Polyomavirus BK (BKV) DNA

Quantified BKV genomic DNA (Advanced Biotechnologies, Inc.) was dilutedin TE buffer to 1, 10, 100, 1,000, 10,000, and 100,000 copy numbers/perreaction. The diluted DNA samples were amplified by Simplexa BKVreal-time PCR assay on an ABI7500 system (Applied Biosystems, Inc.).

The Simplexa BKV real-time PCR assay used a primer mix containing BKVand IC primer pairs. The concentration is 3000 nM for BKV primer pairand 1500 nM for IC primer pair in the primer mix. The sequences of theprimer were listed in the Table 9.

TABLE 9 BKV and IC sequences BK4TQuencher-AGCTGCTATAGGCCTAACTCCTCAAACAT- Scorpion™ ATAGCAGCT(SEQ ID NO: 10) -FAM-AATAGCCCCAGGAGCACCA (SEQ ID NO: 11) BK4-3PCTGTAGAGGGCATAACAAGTACCTCA Reverse (SEQ ID NO: 12) IC Dx2Quencher-TGCGAACTGGCAAGCT Scorpion™ (SEQ ID NO: 4)-Q670-ATTCGCCCTTTGTTTCGACCTA (SEQ ID NO: 5) IC ReverseCCGACGACTGACGAGCAA (SEQ ID NO: 6)

PCR reactions consisted of the same as described in Example 2.

Amplification was carried out by the same method and instrument asdescribed in Example 2. BKV signal (FAM) was collected from channel A,and IC signal (Q670) was collected from channel E on the ABI7500real-time PCR instrument. Signal intensity data was collected for eightreplicates of each diluted DNA sample.

The dilution series was used to demonstrate the linearity of amounts ofinitial target NA estimated by the P/B Method (from s₀). For comparison,signal intensity data was also analyzed by the Threshold Method.

A comparison of the accuracy and reproducibility of Threshold Method andthe P/B Method (from s₀) for samples with 10,000 copy numbers/μl isfound in Table 10.

TABLE 10 Quantitation of BKV at 10,000 copies/μl Threshold P/B Method n= 8 Method (from s_(o)) Mean 8,458 9,035 SD 1937 1303 CV 22.9 14.4 (%)

The linearities of amount of initial target NA estimated by theThreshold Method and the P/B Method (from s₀) for BKV were verycomparable to that demonstrated in Example 2.

Example 5 Quantitation for RNAse P DNA and Quantitation Reproducibility

A RNAse P DNA test plate (an instrument calibration plate provided byABI, P/N 4350583) was used for this study. The test plate includes 20pre-diluted RNAse P target DNA standards and 36 replicates each of 5,000and 10,000 copies/per reaction. The DNA was amplified by ABI's TaqManchemistry included in the test plate, by using default thermal program(50° C. for 2 min for one cycle, 95° C. for 10 min for one cycle, 95° C.for 30 sec for one cycle and 60° C. for 60 sec for 45 cycles) in theABI7500 real-time PCR instrument. Signal intensity data were analyzed bythe Threshold Method and the P/B Method (from s₀).

A comparison of the accuracy and reproducibility of the Threshold Methodand the P/B Method (from s₀) for samples with 5,000 copy numbers/μl isfound in Table 11, and a comparison for samples with 10,000 copynumbers/μl is found in Table 12.

TABLE 11 Quantitation of RNAse at 5,000 copies/μl Threshold P/B Method n= 36 Method (from s_(o)) Mean 4,912 5,082 SD 255 228 CV 5.19 4.50 (%)

TABLE 12 Quantitation of RNAse at 10,000 copies/μl Threshold P/B Methodn = 8 Method (from s_(o)) Mean 10,216 10,182 SD 364 336 CV 3.56 3.30 (%)

The linearities of amount of initial target NA estimated by theThreshold Method and the P/B Method (from s₀) for RNAse P DNA were verycomparable to that demonstrated in Example 2.

Example 6 Quantitation for Herpes Simplex Virus Types 1 and 2 (HSV-1 and-2) in Clinical Samples

A panel of clinical samples (n=57) submitted for HSV1 and 2 testing atFocus Reference Laboratory was tested by Simplexa HSV1 &2 assay on anABI7500 system (Applied Biosystems, Inc.). Genomic DNA was isolated fromthe samples using an automated extraction instrument, MagNA Pure LCsystem (Roche) and MagNA Pure LC Total Nucleic Acid Isolation kit(Roche).

The Simplexa HSV1&2 assay used a primer mix containing HSV-1, HSV-2 andIC primer pairs. The concentration is 3000 nM for HSV-1 primer pair,4000 nM for HSV-2 primer pair, and 1000 nM for IC primer pair in theprimer mix. The sequences of the primer were listed in the Table 13.

TABLE 13 HSV-1, HSV-2 and IC sequences HSV-1Quencher-AGCGGC CTC CGG GTG CCC GGC CA Scorpion™ GCCGCT (SEQ ID NO: 13)-JOE-GAG GAC GAG CTG GCC TTT C (SEQ ID NO: 14) HSV-2-FP-Quencher-ACGCGCGTC TTC CGG GCG TTC CGC D2 GACC GCGCGT Scorpion™(SEQ ID NO: 15) -FAM-GAG GAC GAG CTG GCC TTT C (SEQ ID NO: 16) HSV-1/2GGT GGT GGA CAG GTC GTA GAG RP1 reverse (SEQ ID NO: 17) IC Dx2Quencher-TGCGAACTGGCAAGCT Scorpion™ (SEQ ID NO: 4)-CFR610-ATTCGCCCTTTGTTTCGACCTA (SEQ ID NO: 5) IC ReverseCCGACGACTGACGAGCAA (SEQ ID NO: 6)

PCR reactions consisted of the same as described in Example 2.

Amplification was carried out by the same method and instrument asdescribed in Example 2. HSV-1 signal (JOE) was collected from channel B,HSV-2 signal (FAM) was collected from channel A and IC signal (CFR610)was collected from channel D of the ABI7500 real-time PCR instrument.Signal intensity data was collected for each sample and used to estimatethe amounts of initial target NA by the Threshold Method, the P/B Method(from s₀), and the P/C Method.

Before quantitative analysis, qualitative results were analyzed by theThreshold Method and the P/B Method (from s₀). The positive and negativeagreement of the two methods was compared (see Table 14). Each method,when used for qualitative determination, demonstrated 100% concordancefor all HSV samples.

TABLE 14 Qualitative HSV-1 and -2 Result Comparison P/B Method ThresholdMethod (from s_(o)) positive Negative sum HSV-1 (n = 57) positive 24 024 negative 0 33 33 sum 24 33 57 HSV-2 (n = 57) positive 25 0 25negative 0 32 32 sum 25 32 57

Samples that qualitatively tested positive for HSV-1 (n=24) and HSV-2(n=25) were further analyzed quantitatively by the Threshold Method, theP/B Method (from s₀), and the P/C Method, Comparative data for the threeestimation methods are shown in Tables 15 and 16 for HSV1 and HSV2,respectively. FIGS. 7 and 8 illustrate the correlation of quantitationby the Threshold Method, and the P/B Method (from s₀), respectively,

TABLE 15 Estimated amount of initial target NA in HSV1 Samples ThresholdP/B Method Sample Method (from s_(o)) P/C Method H1-1 9 13 11 H1-221,985 28293 23140 H1-3 45 53 56 H1-4 2,207,380 2977287 2259892 H1-5 1217 13 H1-6 9 15 10 H1-7 2,897 2765 3056 H1-8 257,466 282202 270287 H1-9284,656 327870 276714 H1-10 44,800 56423 46445 H1-11 1,589,580 20656541734360 H1-12 2,421,110 3245448 2598436 H1-13 1,501 1850 1502 H1-14 4744 50 H1-15 1,265,970 1697347 1332712 H1-16 67,998 72651 70177 H1-17823,979 970924 843479 H1-18 212,986 255704 229857 H1-19 36,790 3738536578 H1-20 3 2 3 H1-21 3,529,360 4995628 3926455 H1-22 42,628 5597549484 H1-23 174,049 195894 173026 H1-24 5,201 6154 5878

TABLE 10 Estimated amount of initial target NA in HSV1 Samples ThresholdP/B Method Sample Method (from s_(o)) P/C Method H1-1 3 100 159 H2-1 85157 122 H2-2 90 222 107 H2-3 12 29 15 H2-4 618 811 699 H2-5 33 189 53H2-6 19 26 20 H2-7 8,532 8342 8631 H2-8 293,510 327443 327574 H2-9 2,3322814 2578 H2-10 17,420 17105 17171 H2-11 2,664,040 2762575 2913078 H2-128,201 8321 8605 H2-13 181,092 192221 195365 H2-14 238,659 264466 256857H2-15 17,142 16888 18053 H2-16 641 733 691 H2-17 1,277,940 14059021442496 H2-18 439,061 467888 485901 H2-19 2,544,640 2594372 2768014H2-20 231,880 195453 224689 H2-21 83 127 92 H2-22 89 132 119 H2-23 1,1351083 1226 H2-24 104 157 128

The contents of the articles, patents, and patent applications, and allother documents and electronically available information mentioned orcited herein, are hereby incorporated by reference in their entirety tothe same extent as if each individual publication was specifically andindividually indicated to be incorporated by reference. Applicantsreserve the right to physically incorporate into this application anyand all materials and information from any such articles, patents,patent applications, or other physical and electronic documents.

The methods illustratively described herein may suitably be practiced inthe absence of any element or elements, limitation or limitations, notspecifically disclosed herein. Thus, for example, the terms“comprising”, “including,” containing”, etc. shall be read expansivelyand without limitation. Additionally, the terms and expressions employedherein have been used as terms of description and not of limitation, andthere is no intention in the use of such terms and expressions ofexcluding any equivalents of the features shown and described orportions thereof. It is recognized that various modifications arepossible within the scope of the invention claimed. Thus, it should beunderstood that although the present invention has been specificallydisclosed by preferred embodiments and optional features, modificationand variation of the invention embodied therein herein disclosed may beresorted to by those skilled in the art, and that such modifications andvariations are considered to be within the scope of this invention.

The invention has been described broadly and generically herein. Each ofthe narrower species and subgeneric groupings falling within the genericdisclosure also form part of the methods. This includes the genericdescription of the methods with a proviso or negative limitationremoving any subject matter from the genus, regardless of whether or notthe excised material is specifically recited herein.

1. A method of estimating the initial amount of a target nucleic acid ina sample prior to nucleic acid amplification by polymerase chainreaction (PCR), said method comprising: i) performing PCR on saidsample; ii) obtaining signal intensity data from the performed PCRacross a range of PCR cycles; iii) modeling the signal intensity datawith a phenomenological model and a chemical model, thereby estimatingthe initial amount of target nucleic acid in the sample; and iv)outputting the estimate of the initial amount of target nucleic acid inthe sample to a user or computer readable format; wherein said chemicalmodel is selected from the group consisting of: a)P₃:=s²E³−(s²+(1−D+2q)s)E²+q(2s+q+1)E−q²=0, wherein: S=K₂[S]_(T),${D = \frac{K_{4}}{{K_{2}( {K_{1}\lbrack p\rbrack}_{T} )}^{2}}},$and [S]_(T)=[s]+[h]+[r]+[d]; P₄:=S³E⁴− (S³+(p+2κq+2κ)s²)E³+((p+2κq+κ)+((2κq+κ)p+κ+(−κq−κ)²)s)E²− ((2κqp+(κq+κ)κq)s+(κq+κ)κqp+κ²q)E+κ²q²p=0, wherein: s=K₁[s]_(T), κ=K₁/K₂, and [S]_(T)=[s]+[h]+[r]; and C)P₂:=sE²−(s+q+1)E+q=0, wherein: s=K₂[s]_(T), and [s]_(T)=[s]+[h]+[r];wherein in all of the above formulae: E is the amplification efficiency,p=K₁[p]_(T), q=K₂[q]_(T), [p]_(T)=[p]+[h]+[r], [q]_(T)=[q]+[r],${K_{1} = \frac{\lbrack h_{0} \rbrack}{\lbrack s\rbrack\lbrack p\rbrack}},{K_{2} = \frac{\lbrack r_{0} \rbrack}{\lbrack q\rbrack\lbrack h_{0} \rbrack}},{K_{4} = {\frac{\lbrack d\rbrack}{\lbrack s\rbrack\lbrack s^{*} \rbrack} = \frac{\lbrack d\rbrack}{\lbrack s\rbrack^{2}}}},$[s] denotes the equilibrium concentration of single stranded nucleicacid, [p] denotes the equilibrium concentration of primer, [q] denotesthe equilibrium concentration of polymerase, [h] denotes the equilibriumconcentration of primer-template duplex, with subscripts on h indicatingthe number of nucleotides extending the primer and no subscriptsindicating a fully extended primer, [r] denotes the equilibriumconcentration of reaction complex, with subscripts on r indicating thenumber of nucleotides extending the primer and no subscripts indicatinga fully extended primer, [s*] denotes the equilibrium concentration ofsingle stranded nucleic acid complementary to s, and [d] denotes theequilibrium concentration of double-stranded nucleic acid.
 2. The methodof claim 1, wherein said modeling further comprises estimating theamount of target nucleic acid generated in a reaction mixture in atleast two PCR cycles.
 3. The method of claim 1, wherein said chemicalmodel is the formula recited in part (a) of claim
 1. 4. The method ofclaim 3, further comprising estimating the initial amount of targetnucleic acid from the initial reduced single stranded nucleic acidconcentration, s₀, by the formula ${s_{0} = {x_{0}\frac{K_{2}}{NV}}},$wherein x₀ is the initial amount of target nucleic acid in the sample, Nis Avogadro's number, and V is the sample volume.
 5. The method of claim1, wherein said chemical model is the formula recited in part (b) ofclaim
 1. 6. The method of claim 5, further comprising estimating theinitial amount of target nucleic acid from the initial reduced singlestranded nucleic acid concentration, s₀, by the formula${s_{0} = {x_{0}\frac{K_{1}}{N\; V}}},$ wherein x₀ is the initial amountof target nucleic acid in the sample, N is Avogadro's number, and V isthe sample volume.
 7. The method of claim 1, wherein said chemical modelis the formula recited in part (c) of claim
 1. 8. The method of claim 7,further comprising estimating the initial amount of target nucleic acidfrom the initial reduced single stranded nucleic acid concentration, s₀,by the formula ${s_{0} = {x_{0}\frac{K_{2}}{N\; V}}},$ wherein x₀ is theinitial amount of target nucleic acid in the sample, N is Avogadro'snumber, and V is the sample volume.
 9. The method of claim 1, whereinmodeling the signal intensity with a phenomenological model comprises:v) estimating an efficiency E₀ using the formula${E_{0} = {\max\lbrack {\frac{y_{k - 1} + y_{k} + y_{k + 1}}{y_{k - 2} + y_{k - 1} + y_{k}} - 1} \rbrack}},$wherein y_(k) is signal intensity in cycle number k, vi) estimating aquantity Gx₀ using the formula Gx₀=y_(k) _(max) (1+E₀)^(−k) ^(max) ,wherein E₀ is the efficiency estimated in step (v) and k_(max) is thecycle number k in which E₀ is maximized in step (v), vii) estimating acycle-dependent efficiency E_(i), a background offset a, and a driftconstant b using the phenomenological model${z_{k} = {a + {bk} + {{Gx}_{0}{\prod\limits_{i = 1}^{k}\;( {1 + E_{i}} )}}}},$wherein k is the cycle number, z_(k) is the signal intensity at cycle k,and Gx₀ has the value as estimated in step (vi).
 10. The method of claim1, wherein said modeling comprises using a nonlinear least squares curvefitting approximation method for estimating the parameters of themodels.
 11. The method of claim 10, wherein the nonlinear least squarescurve fitting approximation method is a Levenberg-Marquardtapproximation method.
 12. The method of claim 1, further comprisingidentifying a subset of signal intensity data generated across a rangeof PCR cycles for modeling.
 13. The method of claim 12, whereinidentifying said subset comprises identifying a range of PCR cyclesbeginning before replication is apparent and ending at a cycle where theamplification efficiency, E_(k), has decreased to a predeterminedabsolute lower limit or a relative amount from the initial amplificationefficiency, E₁.
 14. The method of claim 13, wherein the ending cycle isa cycle having an absolute amplification efficiency between 10% and 50%.15. The method of claim 14, wherein the subset comprises the endingcycle and the preceding five to fifteen cycles.
 16. A method ofestimating the initial amount of a target nucleic acid in a sample priorto nucleic acid amplification by polymerase chain reaction, said methodcomprising: i) performing PCR on said sample; ii) obtaining signalintensity data from the performed PCR across a range of PCR cycles; iii)modeling the signal intensity data with a phenomenological model and achemical model, thereby estimating the initial amount of target nucleicacid in the sample; and iv) outputting the estimate of the initialamount of target nucleic acid in the sample to a user or computerreadable format; wherein modeling the signal intensity with aphenomenological model comprises: v) estimating an efficiency E₀ usingthe formula${E_{0} = {\max\lbrack {\frac{y_{k - 1} + y_{k} + y_{k + 1}}{y_{k - 2} + y_{k - 1} + y_{k}} - 1} \rbrack}},$wherein y_(k) is signal intensity in cycle number k, vi) estimating aquantity Gx₀ using the formula Gx₀=y_(k) _(max) (1+E₀)^(−k) ^(max) ,wherein E₀ is the efficiency estimated in step (v) and k_(max) is thecycle number k in which E₀ is maximized in step (v), vii) estimating acycle-dependent efficiency E_(i), a background offset a, and a driftconstant b using the phenomenological model${z_{k} = {a + {b\; k} + {G\; x_{0}{\prod\limits_{i = 1}^{k}\;( {1 + E_{i}} )}}}},$wherein k is the cycle number, z_(k) is the signal intensity at cycle k,and Gx₀ has the value as estimated in step (vi).
 17. The method of claim16, wherein said chemical model is selected from the group consistingof: a) P₃:=s²E³−(s²+(1−D+2q)s)E²+q(2s+q+1)E−q²=0, wherein: S=K₂[S]_(T),${D = \frac{K_{4}}{{K_{2}( {K_{1}\lbrack p\rbrack}_{T} )}^{2}}},$and [S]_(T)=[s]+[h]+[r]+[d]; P₄:=S³E⁴− (S³+(p+2κq+2κ)s²)E³+((p+2κq+κ)+((2κq+κ)p+κ+(−κq−κ)²)s)E²− ((2κqp+(κq+κ)κq)s+(κq+κ)κqp+κ²q)E+κ²q²p=0, wherein: s=K₁[s]_(T), κ=K₁/K₂, and [S]_(T)=[s]+[h]+[r]; and C)P₂:=sE²−(s+q+1)E+q=0, wherein: s=K₂[s]_(T), and [s]_(T)=[s]+[h]+[r];wherein in all of the above formulae: E is the amplification efficiency,p=K₁[p]_(T), q=K₂[q]_(T), [p]_(T)=[p]+[h]+[r], [q]_(T)=[q]+[r],${K_{1} = \frac{\lbrack h_{0} \rbrack}{\lbrack s\rbrack\lbrack p\rbrack}},$[s] denotes the equilibrium concentration of single stranded nucleicacid, [p] denotes the equilibrium concentration of primer, [q] denotesthe equilibrium concentration of polymerase, [h] denotes the equilibriumconcentration of primer-template duplex, with subscripts on h indicatingthe number of nucleotides extending the primer and no subscriptsindicating a fully extended primer, [r] denotes the equilibriumconcentration of reaction complex, with subscripts on r indicating thenumber of nucleotides extending the primer and no subscripts indicatinga fully extended primer, [s*] denotes the equilibrium concentration ofsingle stranded nucleic acid complementary to s, and [d] denotes theequilibrium concentration of double-stranded nucleic acid.
 18. Acomputer program product embodied on a non-transitory computer-readablemedium, the computer program product comprising: (i) computer code thatcauses a processor to receive a signal intensity indicative of an amountof target nucleic acid present in a sample at a multiple times during aPCR amplification; (ii) computer code that causes a processor toestimate the initial target nucleic acid in the sample using aphenomenological model and a chemical model; and (iii) computer codethat causes a processor to output the estimate of initial target nucleicacid in the sample to a user or computer readable media; wherein thecomputer code for estimating the initial target nucleic acid in thesample using a phenomenological model comprises: iv) computer code thatcauses a processor to estimate an efficiency E₀ using the formula${E_{0} = {\max\lbrack {\frac{y_{k - 1} + y_{k} + y_{k + 1}}{y_{k - 2} + y_{k - 1} + y_{k}} - 1} \rbrack}},$wherein y_(k) is signal intensity in cycle number k, v) computer codethat causes a processor to estimate a quantity Gx₀ using the formulaGx₀=y_(k) _(max) (1+E₀)^(−k) ^(max) , wherein E₀ is the efficiencyestimated by the code of (iv) and k_(max) is the cycle number k in whichE₀ is maximized by the code of (iv), vi) computer code that causes aprocessor to estimate a cycle-dependent efficiency E_(i), a backgroundoffset a, and a drift constant b using the phenomenological model${z_{k} = {a + {b\; k} + {G\; x_{0}{\prod\limits_{i = 1}^{k}\;( {1 + E_{i}} )}}}},$wherein k is the cycle number, z_(k) is the signal intensity at cycle k,and Gx₀ has the value as estimated in step (v); and wherein saidchemical model is selected from the group consisting of: a)P₃:=s²E³−(s²+(1−D+2q)s)E²+q(2s+q+1)E−q²=0, wherein: S=K₂[S]_(T),${D = \frac{K_{4}}{{K_{2}( {K_{1}\lbrack p\rbrack}_{T} )}^{2}}},$and [S]_(T)=[s]+[h]+[r]+[d]; P₄:=S³E⁴− (S³+(p+2κq+2κ)s²)E³+((p+2κq+κ)+((2κq+κ)p+κ+(−κq−κ)²)s)E²− ((2κqp+(κq+κ)κq)s+(κq+κ)κqp+κ²q)E+κ²q²p=0, wherein: s=K₁[s]_(T), κ=K₁/K₂, and [S]_(T)=[s]+[h]+[r]; and C)P₂:=sE²−(s+q+1)E+q=0, wherein: s=K₂[s]_(T), and [s]_(T)=[s]+[h]+[r];wherein in all of the above formulae: E is the amplification efficiency,p=K₁[p]_(T), q=K₂[q]_(T), [p]_(T)=[p]+[h]+[r], [q]_(T)=[q]+[r],$\begin{matrix}{{K_{1} = \frac{\lbrack h_{0} \rbrack}{\lbrack s\rbrack\lbrack p\rbrack}},} \\{{K_{2} = \frac{\lbrack r_{0} \rbrack}{\lbrack q\rbrack\lbrack h_{0} \rbrack}},} \\{{K_{4} = {\frac{\lbrack d\rbrack}{\lbrack s\rbrack\lbrack s^{*} \rbrack} = \frac{\lbrack d\rbrack}{\lbrack s\rbrack^{2}}}},}\end{matrix}$ [s] denotes the equilibrium concentration of singlestranded nucleic acid, [p] denotes the equilibrium concentration ofprimer, [q] denotes the equilibrium concentration of polymerase, [h]denotes the equilibrium concentration of primer-template duplex, withsubscripts on h indicating the number of nucleotides extending theprimer and no subscripts indicating a fully extended primer, [r] denotesthe equilibrium concentration of reaction complex, with subscripts on rindicating the number of nucleotides extending the primer and nosubscripts indicating a fully extended primer, [s*] denotes theequilibrium concentration of single stranded nucleic acid complementaryto s, and [d] denotes the equilibrium concentration of double-strandednucleic acid.
 19. The computer program product of claim 18, wherein saidinput is received from a user.
 20. The computer program product of claim18, wherein said input is received from a device.
 21. An apparatuscomprising a processor, and a memory unit coupled to the processor thatcomprises: (i) computer code that causes the processor to receive asignal intensity indicative of an amount of target nucleic acid presentin a sample at a multiple times during a PCR amplification; (ii)computer code that causes the processor to estimate the initial targetnucleic acid in the sample using a phenomenological model and a chemicalmodel; and (iii) computer code that causes the processor to output theestimate of initial target nucleic acid in the sample to a user orcomputer readable media; wherein the computer code for estimating theinitial target nucleic acid in the sample using a phenomenological modelcomprises: iv) computer code that causes the processor to estimate anefficiency E₀ using the formula${E_{0} = {\max\lbrack {\frac{y_{k - 1} + y_{k} + y_{k + 1}}{y_{k - 2} + y_{k - 1} + y_{k}} - 1} \rbrack}},$wherein y_(k) is signal intensity in cycle number k, v) computer codethat causes the processor to estimate a quantity Gx₀ using the formulaGx₀=y_(k) _(max) (1+E₀)^(−k) ^(max) , wherein E₀ is the efficiencyestimated by the code of (iv) and k_(max) is the cycle number k in whichE₀ is maximized by the code of (iv), vi) computer code that causes theprocessor to estimate a cycle-dependent efficiency E_(i), a backgroundoffset a, and a drift constant b using the phenomenological model${z_{k} = {a + {b\; k} + {G\; x_{0}{\prod\limits_{i = 1}^{k}\;( {1 + E_{i}} )}}}},$wherein k is the cycle number, z_(k) is the signal intensity at cycle k,and Gx₀ has the value as estimated by the code of (v); and wherein saidchemical model is selected from the group consisting of: a)P₃:=s²E³−(s²+(1−D+2q)s)E²+q(2s+q+1)E−q²=0, wherein: S=K₂[S]_(T),${D = \frac{K_{4}}{{K_{2}( {K_{1}\lbrack p\rbrack}_{T} )}^{2}}},$and [S]_(T)=[s]+[h]+[r]+[d]; P₄:=S³E⁴− (S³+(p+2κq+2κ)s²)E³+((p+2κq+κ)+((2κq+κ)p+κ+(−κq−κ)²)s)E²− ((2κqp+(κq+κ)κq)s+(κq+κ)κqp+κ²q)E+κ²q²p=0, wherein: s=K₁[s]_(T), κ=K₁/K₂, and [S]_(T)=[s]+[h]+[r]; and C)P₂:=sE²−(s+q+1)E+q=0, wherein: s=K₂[s]_(T), and [s]_(T)=[s]+[h]+[r];wherein in all of the above formulae: E is the amplification efficiency,p=K₁[p]_(T), q=K₂[q]_(T), [p]_(T)=[p]+[h]+[r], [q]_(T)=[q]+[r],$\begin{matrix}{{K_{1} = \frac{\lbrack h_{0} \rbrack}{\lbrack s\rbrack\lbrack p\rbrack}},} \\{{K_{2} = \frac{\lbrack r_{0} \rbrack}{\lbrack q\rbrack\lbrack h_{0} \rbrack}},} \\{{K_{4} = {\frac{\lbrack d\rbrack}{\lbrack s\rbrack\lbrack s^{*} \rbrack} = \frac{\lbrack d\rbrack}{\lbrack s\rbrack^{2}}}},}\end{matrix}$ [s] denotes the equilibrium concentration of singlestranded nucleic acid, [p] denotes the equilibrium concentration ofprimer, [q] denotes the equilibrium concentration of polymerase, [h]denotes the equilibrium concentration of primer-template duplex, withsubscripts on h indicating the number of nucleotides extending theprimer and no subscripts indicating a fully extended primer, [r] denotesthe equilibrium concentration of reaction complex, with subscripts on rindicating the number of nucleotides extending the primer and nosubscripts indicating a fully extended primer, [s*] denotes theequilibrium concentration of single stranded nucleic acid complementaryto s, and [d] denotes the equilibrium concentration of double-strandednucleic acid.
 22. The apparatus of claim 21, wherein said input isreceived from a user.
 23. The apparatus of claim 21, wherein said inputis received from a device.